id	sid	tid	token	lemma	pos
ejpam-546	1	1	10_xxx_nelsen.dvi	10_xxx_nelsen.dvi	NUM
ejpam-546	1	2	european	european	ADJ
ejpam-546	1	3	journal	journal	PROPN
ejpam-546	1	4	of	of	ADP
ejpam-546	1	5	pure	pure	ADJ
ejpam-546	1	6	and	and	CCONJ
ejpam-546	1	7	applied	apply	VERB
ejpam-546	1	8	mathematics	mathematic	NOUN
ejpam-546	1	9	vol	vol	NOUN
ejpam-546	1	10	.	.	PUNCT
ejpam-546	2	1	3	3	NUM
ejpam-546	2	2	,	,	PUNCT
ejpam-546	2	3	no	no	INTJ
ejpam-546	2	4	.	.	NOUN
ejpam-546	2	5	1	1	NUM
ejpam-546	2	6	,	,	PUNCT
ejpam-546	2	7	2010	2010	NUM
ejpam-546	2	8	,	,	PUNCT
ejpam-546	2	9	118	118	NUM
ejpam-546	2	10	-	-	SYM
ejpam-546	2	11	127	127	NUM
ejpam-546	2	12	issn	issn	PROPN
ejpam-546	2	13	1307	1307	NUM
ejpam-546	2	14	-	-	SYM
ejpam-546	2	15	5543	5543	NUM
ejpam-546	2	16	–	–	PUNCT
ejpam-546	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-546	2	18	an	an	DET
ejpam-546	2	19	invitation	invitation	NOUN
ejpam-546	2	20	to	to	AUX
ejpam-546	2	21	proofs	proof	NOUN
ejpam-546	2	22	without	without	ADP
ejpam-546	2	23	words	word	NOUN
ejpam-546	2	24	claudi	claudi	PROPN
ejpam-546	2	25	alsina1	alsina1	PROPN
ejpam-546	2	26	and	and	CCONJ
ejpam-546	3	1	roger	roger	PROPN
ejpam-546	3	2	b.	b.	PROPN
ejpam-546	3	3	nelsen2∗	nelsen2∗	PROPN
ejpam-546	3	4	1	1	NUM
ejpam-546	3	5	universitat	universitat	PROPN
ejpam-546	3	6	politecnica	politecnica	PROPN
ejpam-546	3	7	de	de	PROPN
ejpam-546	3	8	catalunya	catalunya	PROPN
ejpam-546	3	9	,	,	PUNCT
ejpam-546	3	10	secció	secció	PROPN
ejpam-546	3	11	de	de	PROPN
ejpam-546	3	12	matemàtiques	matemàtiques	PROPN
ejpam-546	3	13	i	i	PROPN
ejpam-546	3	14	informàtica	informàtica	PROPN
ejpam-546	3	15	,	,	PUNCT
ejpam-546	3	16	barcelona	barcelona	PROPN
ejpam-546	3	17	,	,	PUNCT
ejpam-546	3	18	spain	spain	PROPN
ejpam-546	3	19	2	2	PROPN
ejpam-546	3	20	lewis	lewis	PROPN
ejpam-546	3	21	&	&	CCONJ
ejpam-546	3	22	clark	clark	PROPN
ejpam-546	3	23	college	college	PROPN
ejpam-546	3	24	,	,	PUNCT
ejpam-546	3	25	department	department	NOUN
ejpam-546	3	26	of	of	ADP
ejpam-546	3	27	mathematical	mathematical	ADJ
ejpam-546	3	28	sciences	sciences	PROPN
ejpam-546	3	29	,	,	PUNCT
ejpam-546	3	30	portland	portland	PROPN
ejpam-546	3	31	oregon	oregon	PROPN
ejpam-546	3	32	,	,	PUNCT
ejpam-546	3	33	usa	usa	PROPN
ejpam-546	3	34	abstract	abstract	PROPN
ejpam-546	3	35	.	.	PUNCT
ejpam-546	4	1	proofs	proof	NOUN
ejpam-546	4	2	without	without	ADP
ejpam-546	4	3	words	word	NOUN
ejpam-546	4	4	are	be	AUX
ejpam-546	4	5	pictures	picture	NOUN
ejpam-546	4	6	or	or	CCONJ
ejpam-546	4	7	diagrams	diagram	NOUN
ejpam-546	4	8	that	that	PRON
ejpam-546	4	9	help	help	VERB
ejpam-546	4	10	the	the	DET
ejpam-546	4	11	reader	reader	NOUN
ejpam-546	4	12	see	see	VERB
ejpam-546	4	13	why	why	SCONJ
ejpam-546	4	14	a	a	DET
ejpam-546	4	15	particular	particular	ADJ
ejpam-546	4	16	mathematical	mathematical	ADJ
ejpam-546	4	17	statement	statement	NOUN
ejpam-546	4	18	may	may	AUX
ejpam-546	4	19	be	be	AUX
ejpam-546	4	20	true	true	ADJ
ejpam-546	4	21	,	,	PUNCT
ejpam-546	4	22	and	and	CCONJ
ejpam-546	4	23	also	also	ADV
ejpam-546	4	24	see	see	VERB
ejpam-546	4	25	how	how	SCONJ
ejpam-546	4	26	one	one	PRON
ejpam-546	4	27	might	might	AUX
ejpam-546	4	28	begin	begin	VERB
ejpam-546	4	29	to	to	PART
ejpam-546	4	30	go	go	VERB
ejpam-546	4	31	about	about	ADP
ejpam-546	4	32	proving	prove	VERB
ejpam-546	4	33	it	it	PRON
ejpam-546	4	34	true	true	ADJ
ejpam-546	4	35	.	.	PUNCT
ejpam-546	5	1	in	in	ADP
ejpam-546	5	2	some	some	DET
ejpam-546	5	3	instances	instance	NOUN
ejpam-546	5	4	a	a	DET
ejpam-546	5	5	proof	proof	NOUN
ejpam-546	5	6	without	without	ADP
ejpam-546	5	7	words	word	NOUN
ejpam-546	5	8	may	may	AUX
ejpam-546	5	9	include	include	VERB
ejpam-546	5	10	an	an	DET
ejpam-546	5	11	equation	equation	NOUN
ejpam-546	5	12	or	or	CCONJ
ejpam-546	5	13	two	two	NUM
ejpam-546	5	14	to	to	PART
ejpam-546	5	15	guide	guide	VERB
ejpam-546	5	16	the	the	DET
ejpam-546	5	17	reader	reader	NOUN
ejpam-546	5	18	,	,	PUNCT
ejpam-546	5	19	but	but	CCONJ
ejpam-546	5	20	the	the	DET
ejpam-546	5	21	emphasis	emphasis	NOUN
ejpam-546	5	22	is	be	AUX
ejpam-546	5	23	clearly	clearly	ADV
ejpam-546	5	24	on	on	ADP
ejpam-546	5	25	providing	provide	VERB
ejpam-546	5	26	visual	visual	ADJ
ejpam-546	5	27	clues	clue	NOUN
ejpam-546	5	28	to	to	PART
ejpam-546	5	29	stimulate	stimulate	VERB
ejpam-546	5	30	mathematical	mathematical	ADJ
ejpam-546	5	31	thought	thought	NOUN
ejpam-546	5	32	.	.	PUNCT
ejpam-546	6	1	while	while	SCONJ
ejpam-546	6	2	proofs	proof	NOUN
ejpam-546	6	3	without	without	ADP
ejpam-546	6	4	words	word	NOUN
ejpam-546	6	5	can	can	AUX
ejpam-546	6	6	be	be	AUX
ejpam-546	6	7	employed	employ	VERB
ejpam-546	6	8	in	in	ADP
ejpam-546	6	9	many	many	ADJ
ejpam-546	6	10	areas	area	NOUN
ejpam-546	6	11	of	of	ADP
ejpam-546	6	12	mathematics	mathematics	NOUN
ejpam-546	6	13	(	(	PUNCT
ejpam-546	6	14	geometry	geometry	NOUN
ejpam-546	6	15	,	,	PUNCT
ejpam-546	6	16	number	number	NOUN
ejpam-546	6	17	theory	theory	NOUN
ejpam-546	6	18	,	,	PUNCT
ejpam-546	6	19	trigonometry	trigonometry	NOUN
ejpam-546	6	20	,	,	PUNCT
ejpam-546	6	21	calculus	calculus	NOUN
ejpam-546	6	22	,	,	PUNCT
ejpam-546	6	23	inequalities	inequality	NOUN
ejpam-546	6	24	,	,	PUNCT
ejpam-546	6	25	and	and	CCONJ
ejpam-546	6	26	so	so	ADV
ejpam-546	6	27	on	on	ADV
ejpam-546	6	28	)	)	PUNCT
ejpam-546	6	29	in	in	ADP
ejpam-546	6	30	our	our	PRON
ejpam-546	6	31	“	"	PUNCT
ejpam-546	6	32	invitation	invitation	NOUN
ejpam-546	6	33	”	"	PUNCT
ejpam-546	6	34	we	we	PRON
ejpam-546	6	35	examine	examine	VERB
ejpam-546	6	36	only	only	ADV
ejpam-546	6	37	one	one	NUM
ejpam-546	6	38	area	area	NOUN
ejpam-546	6	39	:	:	PUNCT
ejpam-546	6	40	elementary	elementary	ADJ
ejpam-546	6	41	combinatorics	combinatoric	NOUN
ejpam-546	6	42	.	.	PUNCT
ejpam-546	7	1	in	in	ADP
ejpam-546	7	2	this	this	DET
ejpam-546	7	3	article	article	NOUN
ejpam-546	7	4	we	we	PRON
ejpam-546	7	5	use	use	VERB
ejpam-546	7	6	combinatorial	combinatorial	ADJ
ejpam-546	7	7	proof	proof	ADJ
ejpam-546	7	8	methods	method	NOUN
ejpam-546	7	9	based	base	VERB
ejpam-546	7	10	on	on	ADP
ejpam-546	7	11	two	two	NUM
ejpam-546	7	12	simple	simple	ADJ
ejpam-546	7	13	counting	counting	NOUN
ejpam-546	7	14	principles	principle	NOUN
ejpam-546	7	15	(	(	PUNCT
ejpam-546	7	16	the	the	DET
ejpam-546	7	17	fubini	fubini	ADJ
ejpam-546	7	18	principle	principle	NOUN
ejpam-546	7	19	and	and	CCONJ
ejpam-546	7	20	the	the	DET
ejpam-546	7	21	cantor	cantor	PROPN
ejpam-546	7	22	principle	principle	NOUN
ejpam-546	7	23	)	)	PUNCT
ejpam-546	7	24	to	to	PART
ejpam-546	7	25	wordlessly	wordlessly	ADV
ejpam-546	7	26	prove	prove	VERB
ejpam-546	7	27	several	several	ADJ
ejpam-546	7	28	simple	simple	ADJ
ejpam-546	7	29	theorems	theorem	NOUN
ejpam-546	7	30	about	about	ADP
ejpam-546	7	31	the	the	DET
ejpam-546	7	32	natural	natural	ADJ
ejpam-546	7	33	numbers	number	NOUN
ejpam-546	7	34	.	.	PUNCT
ejpam-546	8	1	2000	2000	NUM
ejpam-546	8	2	mathematics	mathematic	NOUN
ejpam-546	8	3	subject	subject	NOUN
ejpam-546	8	4	classifications	classification	NOUN
ejpam-546	8	5	:	:	PUNCT
ejpam-546	8	6	00a05	00a05	X
ejpam-546	8	7	key	key	ADJ
ejpam-546	8	8	words	word	NOUN
ejpam-546	8	9	and	and	CCONJ
ejpam-546	8	10	phrases	phrase	NOUN
ejpam-546	8	11	:	:	PUNCT
ejpam-546	8	12	proofs	proof	NOUN
ejpam-546	8	13	without	without	ADP
ejpam-546	8	14	words	word	NOUN
ejpam-546	8	15	,	,	PUNCT
ejpam-546	8	16	visual	visual	ADJ
ejpam-546	8	17	proofs	proof	NOUN
ejpam-546	8	18	,	,	PUNCT
ejpam-546	8	19	visualization	visualization	NOUN
ejpam-546	8	20	in	in	ADP
ejpam-546	8	21	mathematics	mathematic	NOUN
ejpam-546	8	22	1	1	NUM
ejpam-546	8	23	.	.	PUNCT
ejpam-546	9	1	introduction	introduction	NOUN
ejpam-546	9	2	what	what	PRON
ejpam-546	9	3	are	be	AUX
ejpam-546	9	4	“	"	PUNCT
ejpam-546	9	5	proofs	proof	NOUN
ejpam-546	9	6	without	without	ADP
ejpam-546	9	7	words	word	NOUN
ejpam-546	9	8	”	"	PUNCT
ejpam-546	9	9	?	?	PUNCT
ejpam-546	10	1	as	as	SCONJ
ejpam-546	10	2	you	you	PRON
ejpam-546	10	3	will	will	AUX
ejpam-546	10	4	see	see	VERB
ejpam-546	10	5	from	from	ADP
ejpam-546	10	6	this	this	DET
ejpam-546	10	7	article	article	NOUN
ejpam-546	10	8	,	,	PUNCT
ejpam-546	10	9	the	the	DET
ejpam-546	10	10	question	question	NOUN
ejpam-546	10	11	does	do	AUX
ejpam-546	10	12	not	not	PART
ejpam-546	10	13	have	have	VERB
ejpam-546	10	14	a	a	DET
ejpam-546	10	15	simple	simple	ADJ
ejpam-546	10	16	,	,	PUNCT
ejpam-546	10	17	concise	concise	ADJ
ejpam-546	10	18	answer	answer	NOUN
ejpam-546	10	19	.	.	PUNCT
ejpam-546	11	1	generally	generally	ADV
ejpam-546	11	2	,	,	PUNCT
ejpam-546	11	3	proofs	proof	NOUN
ejpam-546	11	4	without	without	ADP
ejpam-546	11	5	words	word	NOUN
ejpam-546	11	6	are	be	AUX
ejpam-546	11	7	pictures	picture	NOUN
ejpam-546	11	8	or	or	CCONJ
ejpam-546	11	9	diagrams	diagram	NOUN
ejpam-546	11	10	that	that	PRON
ejpam-546	11	11	help	help	VERB
ejpam-546	11	12	the	the	DET
ejpam-546	11	13	reader	reader	NOUN
ejpam-546	11	14	see	see	VERB
ejpam-546	11	15	why	why	SCONJ
ejpam-546	11	16	a	a	DET
ejpam-546	11	17	particular	particular	ADJ
ejpam-546	11	18	mathematical	mathematical	ADJ
ejpam-546	11	19	statement	statement	NOUN
ejpam-546	11	20	may	may	AUX
ejpam-546	11	21	be	be	AUX
ejpam-546	11	22	true	true	ADJ
ejpam-546	11	23	,	,	PUNCT
ejpam-546	11	24	and	and	CCONJ
ejpam-546	11	25	also	also	ADV
ejpam-546	11	26	to	to	PART
ejpam-546	11	27	see	see	VERB
ejpam-546	11	28	how	how	SCONJ
ejpam-546	11	29	one	one	PRON
ejpam-546	11	30	might	might	AUX
ejpam-546	11	31	begin	begin	VERB
ejpam-546	11	32	to	to	PART
ejpam-546	11	33	go	go	VERB
ejpam-546	11	34	about	about	ADP
ejpam-546	11	35	proving	prove	VERB
ejpam-546	11	36	it	it	PRON
ejpam-546	11	37	true	true	ADJ
ejpam-546	11	38	.	.	PUNCT
ejpam-546	12	1	as	as	SCONJ
ejpam-546	12	2	yuri	yuri	PROPN
ejpam-546	12	3	ivanovich	ivanovich	PROPN
ejpam-546	12	4	manin	manin	PROPN
ejpam-546	12	5	said	say	VERB
ejpam-546	12	6	,	,	PUNCT
ejpam-546	12	7	“	"	PUNCT
ejpam-546	12	8	a	a	DET
ejpam-546	12	9	good	good	ADJ
ejpam-546	12	10	proof	proof	NOUN
ejpam-546	12	11	is	be	AUX
ejpam-546	12	12	one	one	NUM
ejpam-546	12	13	that	that	PRON
ejpam-546	12	14	makes	make	VERB
ejpam-546	12	15	us	we	PRON
ejpam-546	12	16	wiser	wise	ADJ
ejpam-546	12	17	,	,	PUNCT
ejpam-546	12	18	”	"	PUNCT
ejpam-546	12	19	a	a	DET
ejpam-546	12	20	sentiment	sentiment	NOUN
ejpam-546	12	21	echoed	echo	VERB
ejpam-546	12	22	by	by	ADP
ejpam-546	12	23	andrew	andrew	PROPN
ejpam-546	12	24	gleason	gleason	PROPN
ejpam-546	12	25	:	:	PUNCT
ejpam-546	12	26	“	"	PUNCT
ejpam-546	12	27	proofs	proof	NOUN
ejpam-546	12	28	really	really	ADV
ejpam-546	12	29	are	be	AUX
ejpam-546	12	30	n’t	not	PART
ejpam-546	12	31	there	there	ADV
ejpam-546	12	32	to	to	PART
ejpam-546	12	33	convince	convince	VERB
ejpam-546	12	34	you	you	PRON
ejpam-546	12	35	that	that	SCONJ
ejpam-546	12	36	something	something	PRON
ejpam-546	12	37	is	be	AUX
ejpam-546	12	38	true	true	ADJ
ejpam-546	12	39	they	they	PRON
ejpam-546	12	40	’re	’re	VERB
ejpam-546	12	41	there	there	ADV
ejpam-546	12	42	to	to	PART
ejpam-546	12	43	show	show	VERB
ejpam-546	12	44	you	you	PRON
ejpam-546	12	45	why	why	SCONJ
ejpam-546	12	46	it	it	PRON
ejpam-546	12	47	is	be	AUX
ejpam-546	12	48	true	true	ADJ
ejpam-546	12	49	.	.	PUNCT
ejpam-546	12	50	”	"	PUNCT
ejpam-546	13	1	proofs	proof	NOUN
ejpam-546	13	2	without	without	ADP
ejpam-546	13	3	words	word	NOUN
ejpam-546	13	4	(	(	PUNCT
ejpam-546	13	5	pwws	pwws	NOUN
ejpam-546	13	6	)	)	PUNCT
ejpam-546	13	7	are	be	AUX
ejpam-546	13	8	regular	regular	ADJ
ejpam-546	13	9	features	feature	NOUN
ejpam-546	13	10	in	in	ADP
ejpam-546	13	11	two	two	NUM
ejpam-546	13	12	journals	journal	NOUN
ejpam-546	13	13	published	publish	VERB
ejpam-546	13	14	by	by	ADP
ejpam-546	13	15	the	the	DET
ejpam-546	13	16	mathematical	mathematical	ADJ
ejpam-546	13	17	association	association	PROPN
ejpam-546	13	18	of	of	ADP
ejpam-546	13	19	america	america	PROPN
ejpam-546	13	20	.	.	PUNCT
ejpam-546	14	1	pwws	pwws	PROPN
ejpam-546	14	2	began	begin	VERB
ejpam-546	14	3	to	to	PART
ejpam-546	14	4	appear	appear	VERB
ejpam-546	14	5	in	in	ADP
ejpam-546	14	6	mathematics	mathematics	PROPN
ejpam-546	14	7	magazine	magazine	NOUN
ejpam-546	14	8	about	about	ADP
ejpam-546	14	9	1975	1975	NUM
ejpam-546	14	10	,	,	PUNCT
ejpam-546	14	11	and	and	CCONJ
ejpam-546	14	12	in	in	ADP
ejpam-546	14	13	the	the	DET
ejpam-546	14	14	college	college	NOUN
ejpam-546	14	15	mathematics	mathematics	PROPN
ejpam-546	14	16	journal	journal	PROPN
ejpam-546	14	17	about	about	ADV
ejpam-546	14	18	ten	ten	NUM
ejpam-546	14	19	years	year	NOUN
ejpam-546	14	20	later	later	ADV
ejpam-546	14	21	.	.	PUNCT
ejpam-546	15	1	many	many	ADJ
ejpam-546	15	2	of	of	ADP
ejpam-546	15	3	these	these	PRON
ejpam-546	15	4	appear	appear	VERB
ejpam-546	15	5	in	in	ADP
ejpam-546	15	6	two	two	NUM
ejpam-546	15	7	collections	collection	NOUN
ejpam-546	15	8	of	of	ADP
ejpam-546	15	9	pwws	pwws	NOUN
ejpam-546	15	10	published	publish	VERB
ejpam-546	15	11	by	by	ADP
ejpam-546	15	12	the	the	DET
ejpam-546	15	13	mathematical	mathematical	ADJ
ejpam-546	15	14	association	association	PROPN
ejpam-546	15	15	of	of	ADP
ejpam-546	15	16	america	america	PROPN
ejpam-546	15	17	[	[	X
ejpam-546	15	18	8	8	NUM
ejpam-546	15	19	,	,	PUNCT
ejpam-546	15	20	9	9	NUM
ejpam-546	15	21	]	]	PUNCT
ejpam-546	15	22	.	.	PUNCT
ejpam-546	16	1	∗corresponding	∗corresponde	VERB
ejpam-546	16	2	author	author	NOUN
ejpam-546	16	3	.	.	PUNCT
ejpam-546	17	1	email	email	NOUN
ejpam-546	17	2	address	address	NOUN
ejpam-546	17	3	:	:	PUNCT
ejpam-546	18	1	laudio.alsina	laudio.alsina	NOUN
ejpam-546	18	2	�	�	NOUN
ejpam-546	18	3	up	up	ADP
ejpam-546	18	4	.edu	.edu	PUNCT
ejpam-546	19	1	(	(	PUNCT
ejpam-546	19	2	c.	c.	PROPN
ejpam-546	19	3	alsina	alsina	PROPN
ejpam-546	19	4	)	)	PUNCT
ejpam-546	19	5	,	,	PUNCT
ejpam-546	19	6	nelsen	nelsen	PROPN
ejpam-546	19	7	�	�	PROPN
ejpam-546	19	8	lark.edu	lark.edu	X
ejpam-546	19	9	(	(	PUNCT
ejpam-546	19	10	r.	r.	PROPN
ejpam-546	19	11	nelsen	nelsen	PROPN
ejpam-546	19	12	)	)	PUNCT
ejpam-546	19	13	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-546	20	1	118	118	NUM
ejpam-546	20	2	c	c	NOUN
ejpam-546	20	3	©	©	PROPN
ejpam-546	20	4	2009	2009	NUM
ejpam-546	20	5	ejpam	ejpam	NOUN
ejpam-546	20	6	all	all	DET
ejpam-546	20	7	rights	right	NOUN
ejpam-546	20	8	reserved	reserve	VERB
ejpam-546	20	9	.	.	PUNCT
ejpam-546	21	1	c.	c.	PROPN
ejpam-546	21	2	alsina	alsina	PROPN
ejpam-546	21	3	and	and	CCONJ
ejpam-546	21	4	r.	r.	PROPN
ejpam-546	21	5	nelsen	nelsen	PROPN
ejpam-546	21	6	/	/	SYM
ejpam-546	21	7	eur	eur	PROPN
ejpam-546	21	8	.	.	PUNCT
ejpam-546	22	1	j.	j.	PROPN
ejpam-546	22	2	pure	pure	PROPN
ejpam-546	22	3	appl	appl	PROPN
ejpam-546	22	4	.	.	PROPN
ejpam-546	22	5	math	math	PROPN
ejpam-546	22	6	,	,	PUNCT
ejpam-546	22	7	3	3	NUM
ejpam-546	22	8	(	(	PUNCT
ejpam-546	22	9	2010	2010	NUM
ejpam-546	22	10	)	)	PUNCT
ejpam-546	22	11	,	,	PUNCT
ejpam-546	22	12	118	118	NUM
ejpam-546	22	13	-	-	SYM
ejpam-546	22	14	127	127	NUM
ejpam-546	22	15	119	119	NUM
ejpam-546	22	16	but	but	CCONJ
ejpam-546	22	17	pwws	pwws	NOUN
ejpam-546	22	18	are	be	AUX
ejpam-546	22	19	not	not	PART
ejpam-546	22	20	recent	recent	ADJ
ejpam-546	22	21	innovations	innovation	NOUN
ejpam-546	22	22	-	-	PUNCT
ejpam-546	22	23	they	they	PRON
ejpam-546	22	24	have	have	AUX
ejpam-546	22	25	been	be	AUX
ejpam-546	22	26	around	around	ADV
ejpam-546	22	27	for	for	ADP
ejpam-546	22	28	a	a	DET
ejpam-546	22	29	very	very	ADV
ejpam-546	22	30	long	long	ADJ
ejpam-546	22	31	time	time	NOUN
ejpam-546	22	32	,	,	PUNCT
ejpam-546	22	33	perhaps	perhaps	ADV
ejpam-546	22	34	first	first	ADV
ejpam-546	22	35	appearing	appear	VERB
ejpam-546	22	36	in	in	ADP
ejpam-546	22	37	ancient	ancient	ADJ
ejpam-546	22	38	greece	greece	PROPN
ejpam-546	22	39	and	and	CCONJ
ejpam-546	22	40	china	china	PROPN
ejpam-546	22	41	,	,	PUNCT
ejpam-546	22	42	and	and	CCONJ
ejpam-546	22	43	later	later	ADV
ejpam-546	22	44	in	in	ADP
ejpam-546	22	45	tenth	tenth	ADJ
ejpam-546	22	46	century	century	NOUN
ejpam-546	22	47	arabia	arabia	PROPN
ejpam-546	22	48	and	and	CCONJ
ejpam-546	22	49	renaissance	renaissance	PROPN
ejpam-546	22	50	italy	italy	PROPN
ejpam-546	22	51	.	.	PUNCT
ejpam-546	23	1	today	today	NOUN
ejpam-546	23	2	pwws	pwws	NOUN
ejpam-546	23	3	regularly	regularly	ADV
ejpam-546	23	4	appear	appear	VERB
ejpam-546	23	5	in	in	ADP
ejpam-546	23	6	journals	journal	NOUN
ejpam-546	23	7	published	publish	VERB
ejpam-546	23	8	around	around	ADP
ejpam-546	23	9	the	the	DET
ejpam-546	23	10	world	world	NOUN
ejpam-546	23	11	and	and	CCONJ
ejpam-546	23	12	on	on	ADP
ejpam-546	23	13	the	the	DET
ejpam-546	23	14	world	world	NOUN
ejpam-546	23	15	wide	wide	ADJ
ejpam-546	23	16	web	web	NOUN
ejpam-546	23	17	.	.	PUNCT
ejpam-546	24	1	some	some	PRON
ejpam-546	24	2	argue	argue	VERB
ejpam-546	24	3	that	that	SCONJ
ejpam-546	24	4	pwws	pwws	NOUN
ejpam-546	24	5	are	be	AUX
ejpam-546	24	6	not	not	PART
ejpam-546	24	7	really	really	ADV
ejpam-546	24	8	“	"	PUNCT
ejpam-546	24	9	proofs	proof	NOUN
ejpam-546	24	10	,	,	PUNCT
ejpam-546	24	11	”	"	PUNCT
ejpam-546	24	12	nor	nor	CCONJ
ejpam-546	24	13	,	,	PUNCT
ejpam-546	24	14	for	for	ADP
ejpam-546	24	15	that	that	DET
ejpam-546	24	16	matter	matter	NOUN
ejpam-546	24	17	,	,	PUNCT
ejpam-546	24	18	are	be	AUX
ejpam-546	24	19	they	they	PRON
ejpam-546	24	20	“	"	PUNCT
ejpam-546	24	21	without	without	ADP
ejpam-546	24	22	words	word	NOUN
ejpam-546	24	23	,	,	PUNCT
ejpam-546	24	24	”	"	PUNCT
ejpam-546	24	25	on	on	ADP
ejpam-546	24	26	account	account	NOUN
ejpam-546	24	27	of	of	ADP
ejpam-546	24	28	equations	equation	NOUN
ejpam-546	24	29	which	which	PRON
ejpam-546	24	30	often	often	ADV
ejpam-546	24	31	accompany	accompany	VERB
ejpam-546	24	32	a	a	DET
ejpam-546	24	33	pww	pww	NOUN
ejpam-546	24	34	.	.	PUNCT
ejpam-546	25	1	martin	martin	PROPN
ejpam-546	25	2	gardner	gardner	PROPN
ejpam-546	25	3	,	,	PUNCT
ejpam-546	25	4	in	in	ADP
ejpam-546	25	5	his	his	PRON
ejpam-546	25	6	popular	popular	ADJ
ejpam-546	25	7	“	"	PUNCT
ejpam-546	25	8	mathematical	mathematical	ADJ
ejpam-546	25	9	games	game	NOUN
ejpam-546	25	10	”	"	PUNCT
ejpam-546	25	11	column	column	NOUN
ejpam-546	25	12	in	in	ADP
ejpam-546	25	13	the	the	DET
ejpam-546	25	14	october	october	PROPN
ejpam-546	25	15	1973	1973	NUM
ejpam-546	25	16	issue	issue	NOUN
ejpam-546	25	17	of	of	ADP
ejpam-546	25	18	scientific	scientific	ADJ
ejpam-546	25	19	american	american	PROPN
ejpam-546	25	20	,	,	PUNCT
ejpam-546	25	21	discussed	discuss	VERB
ejpam-546	25	22	pwws	pwws	NOUN
ejpam-546	25	23	as	as	SCONJ
ejpam-546	25	24	“	"	PUNCT
ejpam-546	25	25	look	look	AUX
ejpam-546	25	26	see	see	VERB
ejpam-546	25	27	”	"	PUNCT
ejpam-546	25	28	diagrams	diagram	NOUN
ejpam-546	25	29	.	.	PUNCT
ejpam-546	26	1	he	he	PRON
ejpam-546	26	2	said	say	VERB
ejpam-546	26	3	“	"	PUNCT
ejpam-546	26	4	in	in	ADP
ejpam-546	26	5	many	many	ADJ
ejpam-546	26	6	cases	case	NOUN
ejpam-546	26	7	a	a	DET
ejpam-546	26	8	dull	dull	ADJ
ejpam-546	26	9	proof	proof	NOUN
ejpam-546	26	10	can	can	AUX
ejpam-546	26	11	be	be	AUX
ejpam-546	26	12	supplemented	supplement	VERB
ejpam-546	26	13	by	by	ADP
ejpam-546	26	14	a	a	DET
ejpam-546	26	15	geometric	geometric	ADJ
ejpam-546	26	16	analog	analog	NOUN
ejpam-546	26	17	so	so	ADV
ejpam-546	26	18	simple	simple	ADJ
ejpam-546	26	19	and	and	CCONJ
ejpam-546	26	20	beautiful	beautiful	ADJ
ejpam-546	26	21	that	that	SCONJ
ejpam-546	26	22	the	the	DET
ejpam-546	26	23	truth	truth	NOUN
ejpam-546	26	24	of	of	ADP
ejpam-546	26	25	a	a	DET
ejpam-546	26	26	theorem	theorem	NOUN
ejpam-546	26	27	is	be	AUX
ejpam-546	26	28	almost	almost	ADV
ejpam-546	26	29	seen	see	VERB
ejpam-546	26	30	at	at	ADP
ejpam-546	26	31	a	a	DET
ejpam-546	26	32	glance	glance	NOUN
ejpam-546	26	33	.	.	PUNCT
ejpam-546	26	34	”	"	PUNCT
ejpam-546	27	1	it	it	PRON
ejpam-546	27	2	is	be	AUX
ejpam-546	27	3	in	in	ADP
ejpam-546	27	4	that	that	DET
ejpam-546	27	5	spirit	spirit	NOUN
ejpam-546	27	6	that	that	PRON
ejpam-546	27	7	we	we	PRON
ejpam-546	27	8	write	write	VERB
ejpam-546	27	9	this	this	DET
ejpam-546	27	10	“	"	PUNCT
ejpam-546	27	11	invitation	invitation	NOUN
ejpam-546	27	12	”	"	PUNCT
ejpam-546	27	13	to	to	PART
ejpam-546	27	14	pwws	pwws	VERB
ejpam-546	27	15	.	.	PUNCT
ejpam-546	28	1	in	in	ADP
ejpam-546	28	2	some	some	DET
ejpam-546	28	3	instances	instance	NOUN
ejpam-546	28	4	we	we	PRON
ejpam-546	28	5	include	include	VERB
ejpam-546	28	6	an	an	DET
ejpam-546	28	7	equation	equation	NOUN
ejpam-546	28	8	or	or	CCONJ
ejpam-546	28	9	two	two	NUM
ejpam-546	28	10	to	to	PART
ejpam-546	28	11	guide	guide	VERB
ejpam-546	28	12	the	the	DET
ejpam-546	28	13	reader	reader	NOUN
ejpam-546	28	14	,	,	PUNCT
ejpam-546	28	15	but	but	CCONJ
ejpam-546	28	16	the	the	DET
ejpam-546	28	17	emphasis	emphasis	NOUN
ejpam-546	28	18	is	be	AUX
ejpam-546	28	19	clearly	clearly	ADV
ejpam-546	28	20	on	on	ADP
ejpam-546	28	21	providing	provide	VERB
ejpam-546	28	22	visual	visual	ADJ
ejpam-546	28	23	clues	clue	NOUN
ejpam-546	28	24	to	to	PART
ejpam-546	28	25	stimulate	stimulate	VERB
ejpam-546	28	26	mathematical	mathematical	ADJ
ejpam-546	28	27	thought	thought	NOUN
ejpam-546	28	28	.	.	PUNCT
ejpam-546	29	1	we	we	PRON
ejpam-546	29	2	encourage	encourage	VERB
ejpam-546	29	3	the	the	DET
ejpam-546	29	4	reader	reader	NOUN
ejpam-546	29	5	to	to	PART
ejpam-546	29	6	think	think	VERB
ejpam-546	29	7	about	about	ADP
ejpam-546	29	8	how	how	SCONJ
ejpam-546	29	9	the	the	DET
ejpam-546	29	10	picture	picture	NOUN
ejpam-546	29	11	“	"	PUNCT
ejpam-546	29	12	proves	prove	VERB
ejpam-546	29	13	”	"	PUNCT
ejpam-546	29	14	the	the	DET
ejpam-546	29	15	theorem	theorem	NOUN
ejpam-546	29	16	before	before	ADP
ejpam-546	29	17	reading	read	VERB
ejpam-546	29	18	on	on	ADP
ejpam-546	29	19	.	.	PUNCT
ejpam-546	30	1	however	however	ADV
ejpam-546	30	2	,	,	PUNCT
ejpam-546	30	3	in	in	ADP
ejpam-546	30	4	each	each	DET
ejpam-546	30	5	case	case	NOUN
ejpam-546	30	6	we	we	PRON
ejpam-546	30	7	have	have	AUX
ejpam-546	30	8	included	include	VERB
ejpam-546	30	9	a	a	DET
ejpam-546	30	10	short	short	ADJ
ejpam-546	30	11	description	description	NOUN
ejpam-546	30	12	of	of	ADP
ejpam-546	30	13	what	what	PRON
ejpam-546	30	14	we	we	PRON
ejpam-546	30	15	hope	hope	VERB
ejpam-546	30	16	the	the	DET
ejpam-546	30	17	reader	reader	NOUN
ejpam-546	30	18	sees	see	VERB
ejpam-546	30	19	in	in	ADP
ejpam-546	30	20	each	each	DET
ejpam-546	30	21	picture	picture	NOUN
ejpam-546	30	22	.	.	PUNCT
ejpam-546	31	1	we	we	PRON
ejpam-546	31	2	believe	believe	VERB
ejpam-546	31	3	there	there	PRON
ejpam-546	31	4	is	be	VERB
ejpam-546	31	5	a	a	DET
ejpam-546	31	6	role	role	NOUN
ejpam-546	31	7	for	for	ADP
ejpam-546	31	8	pwws	pwws	NOUN
ejpam-546	31	9	in	in	ADP
ejpam-546	31	10	mathematics	mathematics	NOUN
ejpam-546	31	11	classrooms	classroom	NOUN
ejpam-546	31	12	from	from	ADP
ejpam-546	31	13	elementary	elementary	ADJ
ejpam-546	31	14	schools	school	NOUN
ejpam-546	31	15	to	to	ADP
ejpam-546	31	16	universities	university	NOUN
ejpam-546	31	17	.	.	PUNCT
ejpam-546	32	1	the	the	DET
ejpam-546	32	2	ability	ability	NOUN
ejpam-546	32	3	to	to	PART
ejpam-546	32	4	visualize	visualize	VERB
ejpam-546	32	5	is	be	AUX
ejpam-546	32	6	essential	essential	ADJ
ejpam-546	32	7	for	for	ADP
ejpam-546	32	8	success	success	NOUN
ejpam-546	32	9	in	in	ADP
ejpam-546	32	10	mathematics	mathematic	NOUN
ejpam-546	32	11	,	,	PUNCT
ejpam-546	32	12	and	and	CCONJ
ejpam-546	32	13	george	george	PROPN
ejpam-546	32	14	pólya	pólya	PROPN
ejpam-546	32	15	’s	’s	PART
ejpam-546	32	16	“	"	PUNCT
ejpam-546	32	17	draw	draw	VERB
ejpam-546	32	18	a	a	DET
ejpam-546	32	19	figure	figure	NOUN
ejpam-546	32	20	.	.	PUNCT
ejpam-546	32	21	.	.	PUNCT
ejpam-546	32	22	.	.	PUNCT
ejpam-546	33	1	”	"	PUNCT
ejpam-546	33	2	is	be	AUX
ejpam-546	33	3	classic	classic	ADJ
ejpam-546	33	4	pedagogical	pedagogical	ADJ
ejpam-546	33	5	advice	advice	NOUN
ejpam-546	33	6	.	.	PUNCT
ejpam-546	34	1	2	2	X
ejpam-546	34	2	.	.	X
ejpam-546	34	3	combinatorial	combinatorial	ADJ
ejpam-546	34	4	proofs	proof	NOUN
ejpam-546	34	5	pwws	pwws	NOUN
ejpam-546	34	6	can	can	AUX
ejpam-546	34	7	be	be	AUX
ejpam-546	34	8	employed	employ	VERB
ejpam-546	34	9	in	in	ADP
ejpam-546	34	10	many	many	ADJ
ejpam-546	34	11	areas	area	NOUN
ejpam-546	34	12	of	of	ADP
ejpam-546	34	13	mathematics	mathematic	NOUN
ejpam-546	34	14	,	,	PUNCT
ejpam-546	34	15	to	to	PART
ejpam-546	34	16	prove	prove	VERB
ejpam-546	34	17	theorems	theorem	NOUN
ejpam-546	34	18	in	in	ADP
ejpam-546	34	19	geometry	geometry	NOUN
ejpam-546	34	20	,	,	PUNCT
ejpam-546	34	21	number	number	NOUN
ejpam-546	34	22	theory	theory	NOUN
ejpam-546	34	23	,	,	PUNCT
ejpam-546	34	24	trigonometry	trigonometry	NOUN
ejpam-546	34	25	,	,	PUNCT
ejpam-546	34	26	calculus	calculus	NOUN
ejpam-546	34	27	,	,	PUNCT
ejpam-546	34	28	inequalities	inequality	NOUN
ejpam-546	34	29	,	,	PUNCT
ejpam-546	34	30	and	and	CCONJ
ejpam-546	34	31	so	so	ADV
ejpam-546	34	32	on	on	ADV
ejpam-546	34	33	.	.	PUNCT
ejpam-546	35	1	in	in	ADP
ejpam-546	35	2	our	our	PRON
ejpam-546	35	3	“	"	PUNCT
ejpam-546	35	4	invitation	invitation	NOUN
ejpam-546	35	5	”	"	PUNCT
ejpam-546	35	6	we	we	PRON
ejpam-546	35	7	will	will	AUX
ejpam-546	35	8	examine	examine	VERB
ejpam-546	35	9	only	only	ADV
ejpam-546	35	10	one	one	NUM
ejpam-546	35	11	area	area	NOUN
ejpam-546	35	12	:	:	PUNCT
ejpam-546	35	13	elementary	elementary	ADJ
ejpam-546	35	14	combinatorics	combinatoric	NOUN
ejpam-546	35	15	.	.	PUNCT
ejpam-546	36	1	in	in	ADP
ejpam-546	36	2	many	many	ADJ
ejpam-546	36	3	theorems	theorem	NOUN
ejpam-546	36	4	concerning	concern	VERB
ejpam-546	36	5	the	the	DET
ejpam-546	36	6	natural	natural	ADJ
ejpam-546	36	7	numbers	number	NOUN
ejpam-546	36	8	{	{	PUNCT
ejpam-546	36	9	1,2	1,2	NUM
ejpam-546	36	10	,	,	PUNCT
ejpam-546	36	11	.	.	PUNCT
ejpam-546	36	12	.	.	PUNCT
ejpam-546	36	13	.	.	PUNCT
ejpam-546	36	14	}	}	PUNCT
ejpam-546	36	15	,	,	PUNCT
ejpam-546	36	16	insight	insight	NOUN
ejpam-546	36	17	can	can	AUX
ejpam-546	36	18	be	be	AUX
ejpam-546	36	19	gained	gain	VERB
ejpam-546	36	20	by	by	ADP
ejpam-546	36	21	representing	represent	VERB
ejpam-546	36	22	the	the	DET
ejpam-546	36	23	numbers	number	NOUN
ejpam-546	36	24	as	as	ADP
ejpam-546	36	25	sets	set	NOUN
ejpam-546	36	26	of	of	ADP
ejpam-546	36	27	objects	object	NOUN
ejpam-546	36	28	.	.	PUNCT
ejpam-546	37	1	since	since	SCONJ
ejpam-546	37	2	the	the	DET
ejpam-546	37	3	particular	particular	ADJ
ejpam-546	37	4	choice	choice	NOUN
ejpam-546	37	5	of	of	ADP
ejpam-546	37	6	object	object	NOUN
ejpam-546	37	7	is	be	AUX
ejpam-546	37	8	unimportant	unimportant	ADJ
ejpam-546	37	9	,	,	PUNCT
ejpam-546	37	10	in	in	ADP
ejpam-546	37	11	pwws	pwws	NOUN
ejpam-546	37	12	we	we	PRON
ejpam-546	37	13	usually	usually	ADV
ejpam-546	37	14	use	use	VERB
ejpam-546	37	15	dots	dot	NOUN
ejpam-546	37	16	,	,	PUNCT
ejpam-546	37	17	squares	square	NOUN
ejpam-546	37	18	,	,	PUNCT
ejpam-546	37	19	balls	ball	NOUN
ejpam-546	37	20	,	,	PUNCT
ejpam-546	37	21	cubes	cube	NOUN
ejpam-546	37	22	,	,	PUNCT
ejpam-546	37	23	and	and	CCONJ
ejpam-546	37	24	other	other	ADJ
ejpam-546	37	25	easily	easily	ADV
ejpam-546	37	26	drawn	draw	VERB
ejpam-546	37	27	objects	object	NOUN
ejpam-546	37	28	.	.	PUNCT
ejpam-546	38	1	in	in	ADP
ejpam-546	38	2	this	this	DET
ejpam-546	38	3	article	article	NOUN
ejpam-546	38	4	we	we	PRON
ejpam-546	38	5	will	will	AUX
ejpam-546	38	6	use	use	VERB
ejpam-546	38	7	combinatorial	combinatorial	ADJ
ejpam-546	38	8	proof	proof	ADJ
ejpam-546	38	9	methods	method	NOUN
ejpam-546	38	10	based	base	VERB
ejpam-546	38	11	on	on	ADP
ejpam-546	38	12	two	two	NUM
ejpam-546	38	13	simple	simple	ADJ
ejpam-546	38	14	counting	counting	NOUN
ejpam-546	38	15	principles	principle	NOUN
ejpam-546	38	16	that	that	PRON
ejpam-546	38	17	can	can	AUX
ejpam-546	38	18	be	be	AUX
ejpam-546	38	19	applied	apply	VERB
ejpam-546	38	20	to	to	ADP
ejpam-546	38	21	representations	representation	NOUN
ejpam-546	38	22	of	of	ADP
ejpam-546	38	23	natural	natural	ADJ
ejpam-546	38	24	numbers	number	NOUN
ejpam-546	38	25	by	by	ADP
ejpam-546	38	26	sets	set	NOUN
ejpam-546	38	27	of	of	ADP
ejpam-546	38	28	objects	object	NOUN
ejpam-546	38	29	.	.	PUNCT
ejpam-546	39	1	the	the	DET
ejpam-546	39	2	principles	principle	NOUN
ejpam-546	39	3	are	be	AUX
ejpam-546	39	4	:	:	PUNCT
ejpam-546	39	5	1	1	X
ejpam-546	39	6	.	.	X
ejpam-546	40	1	if	if	SCONJ
ejpam-546	40	2	you	you	PRON
ejpam-546	40	3	count	count	VERB
ejpam-546	40	4	the	the	DET
ejpam-546	40	5	objects	object	NOUN
ejpam-546	40	6	in	in	ADP
ejpam-546	40	7	a	a	DET
ejpam-546	40	8	set	set	NOUN
ejpam-546	40	9	in	in	ADP
ejpam-546	40	10	two	two	NUM
ejpam-546	40	11	different	different	ADJ
ejpam-546	40	12	ways	way	NOUN
ejpam-546	40	13	,	,	PUNCT
ejpam-546	40	14	you	you	PRON
ejpam-546	40	15	will	will	AUX
ejpam-546	40	16	obtain	obtain	VERB
ejpam-546	40	17	the	the	DET
ejpam-546	40	18	same	same	ADJ
ejpam-546	40	19	result	result	NOUN
ejpam-546	40	20	;	;	PUNCT
ejpam-546	40	21	and	and	CCONJ
ejpam-546	40	22	2	2	X
ejpam-546	40	23	.	.	X
ejpam-546	40	24	if	if	SCONJ
ejpam-546	40	25	two	two	NUM
ejpam-546	40	26	sets	set	NOUN
ejpam-546	40	27	are	be	AUX
ejpam-546	40	28	in	in	ADP
ejpam-546	40	29	one	one	NUM
ejpam-546	40	30	-	-	PUNCT
ejpam-546	40	31	to	to	ADP
ejpam-546	40	32	-	-	PUNCT
ejpam-546	40	33	one	one	NUM
ejpam-546	40	34	correspondence	correspondence	NOUN
ejpam-546	40	35	,	,	PUNCT
ejpam-546	40	36	then	then	ADV
ejpam-546	40	37	they	they	PRON
ejpam-546	40	38	have	have	VERB
ejpam-546	40	39	the	the	DET
ejpam-546	40	40	same	same	ADJ
ejpam-546	40	41	number	number	NOUN
ejpam-546	40	42	of	of	ADP
ejpam-546	40	43	elements	element	NOUN
ejpam-546	40	44	.	.	PUNCT
ejpam-546	41	1	the	the	DET
ejpam-546	41	2	first	first	ADJ
ejpam-546	41	3	principle	principle	NOUN
ejpam-546	41	4	has	have	AUX
ejpam-546	41	5	been	be	AUX
ejpam-546	41	6	called	call	VERB
ejpam-546	41	7	the	the	DET
ejpam-546	41	8	fubini	fubini	ADJ
ejpam-546	41	9	principle	principle	NOUN
ejpam-546	41	10	[	[	X
ejpam-546	41	11	11	11	NUM
ejpam-546	41	12	]	]	PUNCT
ejpam-546	41	13	after	after	ADP
ejpam-546	41	14	the	the	DET
ejpam-546	41	15	theorem	theorem	NOUN
ejpam-546	41	16	in	in	ADP
ejpam-546	41	17	multivariable	multivariable	ADJ
ejpam-546	41	18	calculus	calculus	NOUN
ejpam-546	41	19	concerning	concern	VERB
ejpam-546	41	20	exchanging	exchange	VERB
ejpam-546	41	21	the	the	DET
ejpam-546	41	22	order	order	NOUN
ejpam-546	41	23	of	of	ADP
ejpam-546	41	24	integration	integration	NOUN
ejpam-546	41	25	in	in	ADP
ejpam-546	41	26	iterated	iterated	ADJ
ejpam-546	41	27	integrals	integral	NOUN
ejpam-546	41	28	.	.	PUNCT
ejpam-546	42	1	we	we	PRON
ejpam-546	42	2	call	call	VERB
ejpam-546	42	3	the	the	DET
ejpam-546	42	4	second	second	NOUN
ejpam-546	42	5	the	the	DET
ejpam-546	42	6	cantor	cantor	PROPN
ejpam-546	42	7	principle	principle	NOUN
ejpam-546	42	8	,	,	PUNCT
ejpam-546	42	9	after	after	ADP
ejpam-546	42	10	georg	georg	NOUN
ejpam-546	42	11	cantor	cantor	NOUN
ejpam-546	42	12	(	(	PUNCT
ejpam-546	42	13	1845	1845	NUM
ejpam-546	42	14	1918	1918	NUM
ejpam-546	42	15	)	)	PUNCT
ejpam-546	42	16	,	,	PUNCT
ejpam-546	42	17	who	who	PRON
ejpam-546	42	18	used	use	VERB
ejpam-546	42	19	it	it	PRON
ejpam-546	42	20	extensively	extensively	ADV
ejpam-546	42	21	in	in	ADP
ejpam-546	42	22	his	his	PRON
ejpam-546	42	23	investigations	investigation	NOUN
ejpam-546	42	24	into	into	ADP
ejpam-546	42	25	the	the	DET
ejpam-546	42	26	cardinality	cardinality	NOUN
ejpam-546	42	27	of	of	ADP
ejpam-546	42	28	infinite	infinite	ADJ
ejpam-546	42	29	sets	set	NOUN
ejpam-546	42	30	.	.	PUNCT
ejpam-546	43	1	the	the	DET
ejpam-546	43	2	two	two	NUM
ejpam-546	43	3	proof	proof	NOUN
ejpam-546	43	4	techniques	technique	NOUN
ejpam-546	43	5	are	be	AUX
ejpam-546	43	6	also	also	ADV
ejpam-546	43	7	known	know	VERB
ejpam-546	43	8	as	as	ADP
ejpam-546	43	9	the	the	DET
ejpam-546	43	10	double	double	ADJ
ejpam-546	43	11	counting	counting	NOUN
ejpam-546	43	12	method	method	NOUN
ejpam-546	43	13	and	and	CCONJ
ejpam-546	43	14	the	the	DET
ejpam-546	43	15	bijection	bijection	NOUN
ejpam-546	43	16	method	method	NOUN
ejpam-546	43	17	,	,	PUNCT
ejpam-546	43	18	respectively	respectively	ADV
ejpam-546	43	19	.	.	PUNCT
ejpam-546	44	1	c.	c.	PROPN
ejpam-546	44	2	alsina	alsina	PROPN
ejpam-546	44	3	and	and	CCONJ
ejpam-546	44	4	r.	r.	PROPN
ejpam-546	44	5	nelsen	nelsen	PROPN
ejpam-546	44	6	/	/	SYM
ejpam-546	44	7	eur	eur	PROPN
ejpam-546	44	8	.	.	PUNCT
ejpam-546	45	1	j.	j.	PROPN
ejpam-546	45	2	pure	pure	PROPN
ejpam-546	45	3	appl	appl	PROPN
ejpam-546	45	4	.	.	PROPN
ejpam-546	45	5	math	math	PROPN
ejpam-546	45	6	,	,	PUNCT
ejpam-546	45	7	3	3	NUM
ejpam-546	45	8	(	(	PUNCT
ejpam-546	45	9	2010	2010	NUM
ejpam-546	45	10	)	)	PUNCT
ejpam-546	45	11	,	,	PUNCT
ejpam-546	45	12	118	118	NUM
ejpam-546	45	13	-	-	SYM
ejpam-546	45	14	127	127	NUM
ejpam-546	45	15	120	120	NUM
ejpam-546	45	16	3	3	NUM
ejpam-546	45	17	.	.	PUNCT
ejpam-546	45	18	figurate	figurate	NOUN
ejpam-546	45	19	numbers	number	NOUN
ejpam-546	45	20	the	the	DET
ejpam-546	45	21	idea	idea	NOUN
ejpam-546	45	22	of	of	ADP
ejpam-546	45	23	representing	represent	VERB
ejpam-546	45	24	a	a	DET
ejpam-546	45	25	number	number	NOUN
ejpam-546	45	26	by	by	ADP
ejpam-546	45	27	a	a	DET
ejpam-546	45	28	set	set	NOUN
ejpam-546	45	29	of	of	ADP
ejpam-546	45	30	objects	object	NOUN
ejpam-546	45	31	(	(	PUNCT
ejpam-546	45	32	perhaps	perhaps	ADV
ejpam-546	45	33	as	as	ADP
ejpam-546	45	34	pebbles	pebble	NOUN
ejpam-546	45	35	on	on	ADP
ejpam-546	45	36	the	the	DET
ejpam-546	45	37	beach	beach	NOUN
ejpam-546	45	38	)	)	PUNCT
ejpam-546	45	39	dates	date	VERB
ejpam-546	45	40	back	back	ADV
ejpam-546	45	41	at	at	ADP
ejpam-546	45	42	least	least	ADJ
ejpam-546	45	43	to	to	ADP
ejpam-546	45	44	the	the	DET
ejpam-546	45	45	ancient	ancient	ADJ
ejpam-546	45	46	greeks	greeks	PROPN
ejpam-546	45	47	.	.	PUNCT
ejpam-546	46	1	when	when	SCONJ
ejpam-546	46	2	that	that	DET
ejpam-546	46	3	representation	representation	NOUN
ejpam-546	46	4	takes	take	VERB
ejpam-546	46	5	the	the	DET
ejpam-546	46	6	shape	shape	NOUN
ejpam-546	46	7	of	of	ADP
ejpam-546	46	8	a	a	DET
ejpam-546	46	9	polygon	polygon	NOUN
ejpam-546	46	10	such	such	ADJ
ejpam-546	46	11	as	as	ADP
ejpam-546	46	12	a	a	DET
ejpam-546	46	13	triangle	triangle	NOUN
ejpam-546	46	14	or	or	CCONJ
ejpam-546	46	15	a	a	DET
ejpam-546	46	16	square	square	NOUN
ejpam-546	46	17	,	,	PUNCT
ejpam-546	46	18	the	the	DET
ejpam-546	46	19	number	number	NOUN
ejpam-546	46	20	is	be	AUX
ejpam-546	46	21	often	often	ADV
ejpam-546	46	22	called	call	VERB
ejpam-546	46	23	a	a	DET
ejpam-546	46	24	figurate	figurate	ADJ
ejpam-546	46	25	number	number	NOUN
ejpam-546	46	26	.	.	PUNCT
ejpam-546	47	1	we	we	PRON
ejpam-546	47	2	begin	begin	VERB
ejpam-546	47	3	with	with	ADP
ejpam-546	47	4	some	some	DET
ejpam-546	47	5	theorems	theorem	NOUN
ejpam-546	47	6	and	and	CCONJ
ejpam-546	47	7	proofs	proof	NOUN
ejpam-546	47	8	about	about	ADP
ejpam-546	47	9	the	the	DET
ejpam-546	47	10	simplest	simple	ADJ
ejpam-546	47	11	figurate	figurate	NOUN
ejpam-546	47	12	numbers	number	NOUN
ejpam-546	47	13	:	:	PUNCT
ejpam-546	47	14	triangular	triangular	NOUN
ejpam-546	47	15	numbers	number	NOUN
ejpam-546	47	16	and	and	CCONJ
ejpam-546	47	17	squares	square	NOUN
ejpam-546	47	18	.	.	PUNCT
ejpam-546	48	1	nearly	nearly	ADV
ejpam-546	48	2	every	every	DET
ejpam-546	48	3	biography	biography	NOUN
ejpam-546	48	4	of	of	ADP
ejpam-546	48	5	the	the	DET
ejpam-546	48	6	great	great	ADJ
ejpam-546	48	7	mathematician	mathematician	ADJ
ejpam-546	48	8	carl	carl	PROPN
ejpam-546	48	9	friedrich	friedrich	PROPN
ejpam-546	48	10	gauss	gauss	PROPN
ejpam-546	48	11	(	(	PUNCT
ejpam-546	48	12	1777	1777	NUM
ejpam-546	48	13	1855	1855	NUM
ejpam-546	48	14	)	)	PUNCT
ejpam-546	48	15	relates	relate	VERB
ejpam-546	48	16	the	the	DET
ejpam-546	48	17	following	follow	VERB
ejpam-546	48	18	story	story	NOUN
ejpam-546	48	19	.	.	PUNCT
ejpam-546	49	1	when	when	SCONJ
ejpam-546	49	2	gauss	gauss	PROPN
ejpam-546	49	3	was	be	AUX
ejpam-546	49	4	about	about	ADV
ejpam-546	49	5	ten	ten	NUM
ejpam-546	49	6	years	year	NOUN
ejpam-546	49	7	old	old	ADJ
ejpam-546	49	8	,	,	PUNCT
ejpam-546	49	9	his	his	PRON
ejpam-546	49	10	arithmetic	arithmetic	ADJ
ejpam-546	49	11	teacher	teacher	NOUN
ejpam-546	49	12	asked	ask	VERB
ejpam-546	49	13	the	the	DET
ejpam-546	49	14	students	student	NOUN
ejpam-546	49	15	in	in	ADP
ejpam-546	49	16	class	class	NOUN
ejpam-546	49	17	to	to	PART
ejpam-546	49	18	compute	compute	VERB
ejpam-546	49	19	the	the	DET
ejpam-546	49	20	sum	sum	NOUN
ejpam-546	49	21	1	1	NUM
ejpam-546	49	22	+	+	NUM
ejpam-546	49	23	2	2	NUM
ejpam-546	49	24	+	+	NUM
ejpam-546	49	25	3	3	NUM
ejpam-546	49	26	+	+	NUM
ejpam-546	49	27	.	.	PUNCT
ejpam-546	49	28	.	.	PUNCT
ejpam-546	50	1	.+	.+	NOUN
ejpam-546	50	2	100	100	NUM
ejpam-546	50	3	,	,	PUNCT
ejpam-546	50	4	anticipating	anticipate	VERB
ejpam-546	50	5	this	this	PRON
ejpam-546	50	6	would	would	AUX
ejpam-546	50	7	keep	keep	VERB
ejpam-546	50	8	the	the	DET
ejpam-546	50	9	students	student	NOUN
ejpam-546	50	10	busy	busy	ADJ
ejpam-546	50	11	for	for	ADP
ejpam-546	50	12	some	some	DET
ejpam-546	50	13	time	time	NOUN
ejpam-546	50	14	.	.	PUNCT
ejpam-546	51	1	he	he	PRON
ejpam-546	51	2	barely	barely	ADV
ejpam-546	51	3	finished	finish	VERB
ejpam-546	51	4	stating	state	VERB
ejpam-546	51	5	the	the	DET
ejpam-546	51	6	problem	problem	NOUN
ejpam-546	51	7	when	when	SCONJ
ejpam-546	51	8	young	young	ADJ
ejpam-546	51	9	carl	carl	PROPN
ejpam-546	51	10	came	come	VERB
ejpam-546	51	11	forward	forward	ADV
ejpam-546	51	12	and	and	CCONJ
ejpam-546	51	13	placed	place	VERB
ejpam-546	51	14	his	his	PRON
ejpam-546	51	15	slate	slate	NOUN
ejpam-546	51	16	on	on	ADP
ejpam-546	51	17	the	the	DET
ejpam-546	51	18	teacher	teacher	NOUN
ejpam-546	51	19	’s	’s	PART
ejpam-546	51	20	desk	desk	NOUN
ejpam-546	51	21	,	,	PUNCT
ejpam-546	51	22	void	void	NOUN
ejpam-546	51	23	of	of	ADP
ejpam-546	51	24	calculation	calculation	NOUN
ejpam-546	51	25	,	,	PUNCT
ejpam-546	51	26	with	with	ADP
ejpam-546	51	27	the	the	DET
ejpam-546	51	28	correct	correct	ADJ
ejpam-546	51	29	answer	answer	NOUN
ejpam-546	51	30	:	:	PUNCT
ejpam-546	51	31	5050	5050	NUM
ejpam-546	51	32	.	.	PUNCT
ejpam-546	52	1	when	when	SCONJ
ejpam-546	52	2	asked	ask	VERB
ejpam-546	52	3	to	to	PART
ejpam-546	52	4	explain	explain	VERB
ejpam-546	52	5	,	,	PUNCT
ejpam-546	52	6	gauss	gauss	PROPN
ejpam-546	52	7	admitted	admit	VERB
ejpam-546	52	8	he	he	PRON
ejpam-546	52	9	recognized	recognize	VERB
ejpam-546	52	10	the	the	DET
ejpam-546	52	11	pattern	pattern	NOUN
ejpam-546	52	12	1	1	NUM
ejpam-546	52	13	+	+	SYM
ejpam-546	53	1	100	100	NUM
ejpam-546	53	2	=	=	SYM
ejpam-546	53	3	101,2	101,2	NUM
ejpam-546	53	4	+	+	CCONJ
ejpam-546	53	5	99=	99=	NUM
ejpam-546	53	6	101,3	101,3	NUM
ejpam-546	53	7	+	+	NUM
ejpam-546	53	8	98=	98=	NUM
ejpam-546	53	9	101	101	NUM
ejpam-546	53	10	,	,	PUNCT
ejpam-546	53	11	and	and	CCONJ
ejpam-546	53	12	so	so	ADV
ejpam-546	53	13	on	on	ADP
ejpam-546	53	14	to	to	ADP
ejpam-546	53	15	50	50	NUM
ejpam-546	53	16	+	+	NOUN
ejpam-546	53	17	51=	51=	NOUN
ejpam-546	53	18	101	101	NUM
ejpam-546	53	19	.	.	PUNCT
ejpam-546	54	1	since	since	SCONJ
ejpam-546	54	2	there	there	PRON
ejpam-546	54	3	are	be	VERB
ejpam-546	54	4	fifty	fifty	NUM
ejpam-546	54	5	such	such	ADJ
ejpam-546	54	6	pairs	pair	NOUN
ejpam-546	54	7	,	,	PUNCT
ejpam-546	54	8	the	the	DET
ejpam-546	54	9	sum	sum	NOUN
ejpam-546	54	10	must	must	AUX
ejpam-546	54	11	be	be	AUX
ejpam-546	54	12	50×	50×	NUM
ejpam-546	54	13	101	101	NUM
ejpam-546	54	14	=	=	SYM
ejpam-546	54	15	5050	5050	NUM
ejpam-546	54	16	.	.	PUNCT
ejpam-546	55	1	the	the	DET
ejpam-546	55	2	pattern	pattern	NOUN
ejpam-546	55	3	for	for	ADP
ejpam-546	55	4	the	the	DET
ejpam-546	55	5	sum	sum	NOUN
ejpam-546	55	6	(	(	PUNCT
ejpam-546	55	7	adding	add	VERB
ejpam-546	55	8	the	the	DET
ejpam-546	55	9	largest	large	ADJ
ejpam-546	55	10	number	number	NOUN
ejpam-546	55	11	to	to	ADP
ejpam-546	55	12	the	the	DET
ejpam-546	55	13	smallest	small	ADJ
ejpam-546	55	14	,	,	PUNCT
ejpam-546	55	15	the	the	DET
ejpam-546	55	16	second	second	ADV
ejpam-546	55	17	largest	large	ADJ
ejpam-546	55	18	to	to	ADP
ejpam-546	55	19	the	the	DET
ejpam-546	55	20	second	second	ADJ
ejpam-546	55	21	smallest	small	ADJ
ejpam-546	55	22	,	,	PUNCT
ejpam-546	55	23	and	and	CCONJ
ejpam-546	55	24	so	so	ADV
ejpam-546	55	25	on	on	ADV
ejpam-546	55	26	)	)	PUNCT
ejpam-546	55	27	is	be	AUX
ejpam-546	55	28	illustrated	illustrate	VERB
ejpam-546	55	29	in	in	ADP
ejpam-546	55	30	figure	figure	NOUN
ejpam-546	55	31	1	1	NUM
ejpam-546	55	32	,	,	PUNCT
ejpam-546	55	33	where	where	SCONJ
ejpam-546	55	34	the	the	DET
ejpam-546	55	35	rows	row	NOUN
ejpam-546	55	36	of	of	ADP
ejpam-546	55	37	balls	ball	NOUN
ejpam-546	55	38	represent	represent	VERB
ejpam-546	55	39	positive	positive	ADJ
ejpam-546	55	40	integers	integer	NOUN
ejpam-546	55	41	.	.	PUNCT
ejpam-546	56	1	figure	figure	VERB
ejpam-546	56	2	1	1	NUM
ejpam-546	56	3	the	the	DET
ejpam-546	56	4	number	number	NOUN
ejpam-546	56	5	tn	tn	NOUN
ejpam-546	57	1	=	=	SYM
ejpam-546	57	2	1	1	NUM
ejpam-546	57	3	+	+	NUM
ejpam-546	57	4	2	2	NUM
ejpam-546	57	5	+	+	NUM
ejpam-546	57	6	3	3	NUM
ejpam-546	57	7	+	+	NUM
ejpam-546	57	8	.	.	PUNCT
ejpam-546	57	9	.	.	PUNCT
ejpam-546	57	10	.	.	PUNCT
ejpam-546	58	1	+	+	CCONJ
ejpam-546	58	2	n	n	CCONJ
ejpam-546	58	3	for	for	ADP
ejpam-546	58	4	a	a	DET
ejpam-546	58	5	positive	positive	ADJ
ejpam-546	58	6	integer	integer	NOUN
ejpam-546	58	7	n	n	NUM
ejpam-546	58	8	is	be	AUX
ejpam-546	58	9	called	call	VERB
ejpam-546	58	10	the	the	DET
ejpam-546	58	11	nth	nth	NOUN
ejpam-546	58	12	triangular	triangular	NOUN
ejpam-546	58	13	number	number	NOUN
ejpam-546	58	14	,	,	PUNCT
ejpam-546	58	15	from	from	ADP
ejpam-546	58	16	the	the	DET
ejpam-546	58	17	pattern	pattern	NOUN
ejpam-546	58	18	of	of	ADP
ejpam-546	58	19	the	the	DET
ejpam-546	58	20	dots	dot	NOUN
ejpam-546	58	21	on	on	ADP
ejpam-546	58	22	the	the	DET
ejpam-546	58	23	left	left	NOUN
ejpam-546	58	24	in	in	ADP
ejpam-546	58	25	figure	figure	NOUN
ejpam-546	58	26	1	1	NUM
ejpam-546	58	27	.	.	PUNCT
ejpam-546	59	1	young	young	ADJ
ejpam-546	59	2	carl	carl	PROPN
ejpam-546	59	3	correctly	correctly	ADV
ejpam-546	59	4	computed	compute	VERB
ejpam-546	59	5	t100	t100	PROPN
ejpam-546	59	6	=	=	SYM
ejpam-546	59	7	5050	5050	NUM
ejpam-546	59	8	.	.	PUNCT
ejpam-546	60	1	however	however	ADV
ejpam-546	60	2	,	,	PUNCT
ejpam-546	60	3	this	this	DET
ejpam-546	60	4	solution	solution	NOUN
ejpam-546	60	5	works	work	VERB
ejpam-546	60	6	only	only	ADV
ejpam-546	60	7	for	for	ADP
ejpam-546	60	8	n	n	PRON
ejpam-546	60	9	even	even	ADV
ejpam-546	60	10	,	,	PUNCT
ejpam-546	60	11	so	so	ADV
ejpam-546	60	12	we	we	PRON
ejpam-546	60	13	first	first	ADV
ejpam-546	60	14	prove	prove	VERB
ejpam-546	60	15	theorem	theorem	VERB
ejpam-546	60	16	1	1	NUM
ejpam-546	60	17	.	.	PUNCT
ejpam-546	60	18	for	for	ADP
ejpam-546	60	19	all	all	DET
ejpam-546	60	20	n≥	n≥	ADJ
ejpam-546	60	21	1	1	NUM
ejpam-546	60	22	,	,	PUNCT
ejpam-546	60	23	tn	tn	NOUN
ejpam-546	60	24	=	=	SYM
ejpam-546	60	25	n(n+1	n(n+1	PROPN
ejpam-546	60	26	)	)	PUNCT
ejpam-546	60	27	2	2	NUM
ejpam-546	60	28	.	.	PUNCT
ejpam-546	61	1	proof	proof	NOUN
ejpam-546	61	2	.	.	PUNCT
ejpam-546	62	1	see	see	VERB
ejpam-546	62	2	figure	figure	NOUN
ejpam-546	62	3	2	2	NUM
ejpam-546	62	4	.	.	PUNCT
ejpam-546	62	5	figure	figure	NOUN
ejpam-546	62	6	2	2	NUM
ejpam-546	62	7	c.	c.	PROPN
ejpam-546	62	8	alsina	alsina	PROPN
ejpam-546	62	9	and	and	CCONJ
ejpam-546	62	10	r.	r.	PROPN
ejpam-546	62	11	nelsen	nelsen	PROPN
ejpam-546	62	12	/	/	SYM
ejpam-546	62	13	eur	eur	PROPN
ejpam-546	62	14	.	.	PUNCT
ejpam-546	63	1	j.	j.	PROPN
ejpam-546	63	2	pure	pure	PROPN
ejpam-546	63	3	appl	appl	PROPN
ejpam-546	63	4	.	.	PROPN
ejpam-546	63	5	math	math	PROPN
ejpam-546	63	6	,	,	PUNCT
ejpam-546	63	7	3	3	NUM
ejpam-546	63	8	(	(	PUNCT
ejpam-546	63	9	2010	2010	NUM
ejpam-546	63	10	)	)	PUNCT
ejpam-546	63	11	,	,	PUNCT
ejpam-546	63	12	118	118	NUM
ejpam-546	63	13	-	-	SYM
ejpam-546	63	14	127	127	NUM
ejpam-546	63	15	121	121	NUM
ejpam-546	63	16	we	we	PRON
ejpam-546	63	17	arrange	arrange	VERB
ejpam-546	63	18	two	two	NUM
ejpam-546	63	19	copies	copy	NOUN
ejpam-546	63	20	of	of	ADP
ejpam-546	63	21	tn	tn	PRON
ejpam-546	63	22	to	to	PART
ejpam-546	63	23	form	form	VERB
ejpam-546	63	24	a	a	DET
ejpam-546	63	25	rectangular	rectangular	ADJ
ejpam-546	63	26	array	array	NOUN
ejpam-546	63	27	of	of	ADP
ejpam-546	63	28	balls	ball	NOUN
ejpam-546	63	29	in	in	ADP
ejpam-546	63	30	n	n	CCONJ
ejpam-546	63	31	rows	row	NOUN
ejpam-546	63	32	and	and	CCONJ
ejpam-546	63	33	n	n	PRON
ejpam-546	63	34	+	+	CCONJ
ejpam-546	63	35	1	1	NUM
ejpam-546	63	36	columns	column	NOUN
ejpam-546	63	37	.	.	PUNCT
ejpam-546	64	1	then	then	ADV
ejpam-546	64	2	we	we	PRON
ejpam-546	64	3	have	have	VERB
ejpam-546	64	4	2tn	2tn	ADJ
ejpam-546	64	5	=	=	SYM
ejpam-546	64	6	n(n+	n(n+	NUM
ejpam-546	64	7	1	1	NUM
ejpam-546	64	8	)	)	PUNCT
ejpam-546	64	9	,	,	PUNCT
ejpam-546	64	10	or	or	CCONJ
ejpam-546	64	11	tn	tn	X
ejpam-546	64	12	=	=	SYM
ejpam-546	64	13	n(n+	n(n+	PROPN
ejpam-546	64	14	1)/2	1)/2	NUM
ejpam-546	64	15	.	.	PUNCT
ejpam-546	65	1	the	the	DET
ejpam-546	65	2	counting	counting	NOUN
ejpam-546	65	3	procedure	procedure	NOUN
ejpam-546	65	4	in	in	ADP
ejpam-546	65	5	the	the	DET
ejpam-546	65	6	preceding	precede	VERB
ejpam-546	65	7	combinatorial	combinatorial	ADJ
ejpam-546	65	8	proof	proof	NOUN
ejpam-546	65	9	is	be	AUX
ejpam-546	65	10	the	the	DET
ejpam-546	65	11	fubini	fubini	ADJ
ejpam-546	65	12	principle	principle	NOUN
ejpam-546	65	13	.	.	PUNCT
ejpam-546	66	1	we	we	PRON
ejpam-546	66	2	employ	employ	VERB
ejpam-546	66	3	the	the	DET
ejpam-546	66	4	same	same	ADJ
ejpam-546	66	5	procedure	procedure	NOUN
ejpam-546	66	6	to	to	PART
ejpam-546	66	7	prove	prove	VERB
ejpam-546	66	8	that	that	SCONJ
ejpam-546	66	9	sums	sum	NOUN
ejpam-546	66	10	of	of	ADP
ejpam-546	66	11	odd	odd	ADJ
ejpam-546	66	12	numbers	number	NOUN
ejpam-546	66	13	are	be	AUX
ejpam-546	66	14	squares	square	NOUN
ejpam-546	66	15	.	.	PUNCT
ejpam-546	67	1	theorem	theorem	NOUN
ejpam-546	67	2	2	2	NUM
ejpam-546	67	3	.	.	X
ejpam-546	67	4	for	for	ADP
ejpam-546	67	5	all	all	DET
ejpam-546	67	6	n≥	n≥	ADJ
ejpam-546	67	7	1	1	NUM
ejpam-546	67	8	,	,	PUNCT
ejpam-546	67	9	1	1	NUM
ejpam-546	67	10	+	+	SYM
ejpam-546	67	11	3	3	NUM
ejpam-546	67	12	+	+	NUM
ejpam-546	67	13	5	5	NUM
ejpam-546	67	14	+	+	NUM
ejpam-546	67	15	.	.	PUNCT
ejpam-546	67	16	.	.	PUNCT
ejpam-546	68	1	.+	.+	NOUN
ejpam-546	68	2	(	(	PUNCT
ejpam-546	68	3	2n−	2n−	NUM
ejpam-546	68	4	1	1	NUM
ejpam-546	68	5	)	)	PUNCT
ejpam-546	68	6	=	=	SYM
ejpam-546	68	7	n2	n2	NOUN
ejpam-546	68	8	.	.	PUNCT
ejpam-546	68	9	proof	proof	NOUN
ejpam-546	68	10	.	.	PUNCT
ejpam-546	69	1	we	we	PRON
ejpam-546	69	2	give	give	VERB
ejpam-546	69	3	two	two	NUM
ejpam-546	69	4	combinatorial	combinatorial	ADJ
ejpam-546	69	5	proofs	proof	NOUN
ejpam-546	69	6	in	in	ADP
ejpam-546	69	7	figure	figure	NOUN
ejpam-546	69	8	3	3	NUM
ejpam-546	69	9	.	.	PUNCT
ejpam-546	70	1	(	(	PUNCT
ejpam-546	70	2	a	a	X
ejpam-546	70	3	)	)	PUNCT
ejpam-546	70	4	(	(	PUNCT
ejpam-546	70	5	b	b	X
ejpam-546	70	6	)	)	PUNCT
ejpam-546	70	7	figure	figure	NOUN
ejpam-546	70	8	3	3	NUM
ejpam-546	70	9	in	in	ADP
ejpam-546	70	10	figure	figure	NOUN
ejpam-546	70	11	3a	3a	NUM
ejpam-546	70	12	,	,	PUNCT
ejpam-546	70	13	we	we	PRON
ejpam-546	70	14	count	count	VERB
ejpam-546	70	15	the	the	DET
ejpam-546	70	16	balls	ball	NOUN
ejpam-546	70	17	in	in	ADP
ejpam-546	70	18	two	two	NUM
ejpam-546	70	19	ways	way	NOUN
ejpam-546	70	20	,	,	PUNCT
ejpam-546	70	21	first	first	ADV
ejpam-546	70	22	as	as	ADP
ejpam-546	70	23	a	a	DET
ejpam-546	70	24	square	square	ADJ
ejpam-546	70	25	array	array	NOUN
ejpam-546	70	26	of	of	ADP
ejpam-546	70	27	balls	ball	NOUN
ejpam-546	70	28	,	,	PUNCT
ejpam-546	70	29	and	and	CCONJ
ejpam-546	70	30	then	then	ADV
ejpam-546	70	31	by	by	ADP
ejpam-546	70	32	the	the	DET
ejpam-546	70	33	number	number	NOUN
ejpam-546	70	34	of	of	ADP
ejpam-546	70	35	balls	ball	NOUN
ejpam-546	70	36	in	in	ADP
ejpam-546	70	37	each	each	DET
ejpam-546	70	38	l	l	NOUN
ejpam-546	70	39	-	-	PUNCT
ejpam-546	70	40	shaped	shape	VERB
ejpam-546	70	41	region	region	NOUN
ejpam-546	70	42	of	of	ADP
ejpam-546	70	43	similarly	similarly	ADV
ejpam-546	70	44	colored	colored	ADJ
ejpam-546	70	45	balls	ball	NOUN
ejpam-546	70	46	(	(	PUNCT
ejpam-546	70	47	the	the	DET
ejpam-546	70	48	fubini	fubini	ADJ
ejpam-546	70	49	principle	principle	NOUN
ejpam-546	70	50	)	)	PUNCT
ejpam-546	70	51	.	.	PUNCT
ejpam-546	71	1	in	in	ADP
ejpam-546	71	2	figure	figure	NOUN
ejpam-546	71	3	3b	3b	NUM
ejpam-546	71	4	,	,	PUNCT
ejpam-546	71	5	we	we	PRON
ejpam-546	71	6	see	see	VERB
ejpam-546	71	7	a	a	DET
ejpam-546	71	8	one	one	NUM
ejpam-546	71	9	-	-	PUNCT
ejpam-546	71	10	to	to	ADP
ejpam-546	71	11	one	one	NUM
ejpam-546	71	12	correspondence	correspondence	NOUN
ejpam-546	71	13	(	(	PUNCT
ejpam-546	71	14	illustrated	illustrate	VERB
ejpam-546	71	15	by	by	ADP
ejpam-546	71	16	the	the	DET
ejpam-546	71	17	color	color	NOUN
ejpam-546	71	18	of	of	ADP
ejpam-546	71	19	the	the	DET
ejpam-546	71	20	balls	ball	NOUN
ejpam-546	71	21	)	)	PUNCT
ejpam-546	71	22	between	between	ADP
ejpam-546	71	23	a	a	DET
ejpam-546	71	24	triangular	triangular	NOUN
ejpam-546	71	25	array	array	NOUN
ejpam-546	71	26	of	of	ADP
ejpam-546	71	27	balls	ball	NOUN
ejpam-546	71	28	in	in	ADP
ejpam-546	71	29	rows	row	NOUN
ejpam-546	71	30	with	with	ADP
ejpam-546	71	31	1,3,5	1,3,5	NUM
ejpam-546	71	32	,	,	PUNCT
ejpam-546	71	33	.	.	PUNCT
ejpam-546	71	34	.	.	PUNCT
ejpam-546	71	35	.	.	PUNCT
ejpam-546	72	1	,	,	PUNCT
ejpam-546	72	2	(	(	PUNCT
ejpam-546	72	3	2n−	2n−	NUM
ejpam-546	72	4	1	1	NUM
ejpam-546	72	5	)	)	PUNCT
ejpam-546	72	6	balls	ball	NOUN
ejpam-546	72	7	,	,	PUNCT
ejpam-546	72	8	and	and	CCONJ
ejpam-546	72	9	a	a	DET
ejpam-546	72	10	square	square	ADJ
ejpam-546	72	11	array	array	NOUN
ejpam-546	72	12	of	of	ADP
ejpam-546	72	13	balls	ball	NOUN
ejpam-546	72	14	(	(	PUNCT
ejpam-546	72	15	the	the	DET
ejpam-546	72	16	cantor	cantor	PROPN
ejpam-546	72	17	principle	principle	PROPN
ejpam-546	72	18	)	)	PUNCT
ejpam-546	72	19	.	.	PUNCT
ejpam-546	73	1	the	the	DET
ejpam-546	73	2	same	same	ADJ
ejpam-546	73	3	idea	idea	NOUN
ejpam-546	73	4	can	can	AUX
ejpam-546	73	5	be	be	AUX
ejpam-546	73	6	employed	employ	VERB
ejpam-546	73	7	in	in	ADP
ejpam-546	73	8	three	three	NUM
ejpam-546	73	9	dimensions	dimension	NOUN
ejpam-546	73	10	to	to	PART
ejpam-546	73	11	establish	establish	VERB
ejpam-546	73	12	the	the	DET
ejpam-546	73	13	following	follow	VERB
ejpam-546	73	14	sequence	sequence	NOUN
ejpam-546	73	15	of	of	ADP
ejpam-546	73	16	identities	identity	NOUN
ejpam-546	73	17	:	:	PUNCT
ejpam-546	73	18	1	1	NUM
ejpam-546	73	19	+	+	NUM
ejpam-546	73	20	2=	2=	NUM
ejpam-546	73	21	3	3	NUM
ejpam-546	73	22	,	,	PUNCT
ejpam-546	73	23	4	4	NUM
ejpam-546	73	24	+	+	SYM
ejpam-546	73	25	5	5	NUM
ejpam-546	73	26	+	+	CCONJ
ejpam-546	73	27	6=	6=	ADP
ejpam-546	73	28	7	7	NUM
ejpam-546	73	29	+	+	NUM
ejpam-546	73	30	8	8	NUM
ejpam-546	73	31	,	,	PUNCT
ejpam-546	73	32	9	9	NUM
ejpam-546	73	33	+	+	SYM
ejpam-546	73	34	10	10	NUM
ejpam-546	73	35	+	+	NUM
ejpam-546	73	36	11	11	NUM
ejpam-546	73	37	+	+	NOUN
ejpam-546	73	38	12=	12=	NUM
ejpam-546	73	39	13	13	NUM
ejpam-546	73	40	+	+	NUM
ejpam-546	73	41	14	14	NUM
ejpam-546	73	42	+	+	SYM
ejpam-546	73	43	15	15	NUM
ejpam-546	73	44	,	,	PUNCT
ejpam-546	73	45	etc	etc	X
ejpam-546	73	46	.	.	X
ejpam-546	73	47	note	note	VERB
ejpam-546	73	48	that	that	SCONJ
ejpam-546	73	49	each	each	DET
ejpam-546	73	50	row	row	NOUN
ejpam-546	73	51	begins	begin	VERB
ejpam-546	73	52	with	with	ADP
ejpam-546	73	53	a	a	DET
ejpam-546	73	54	square	square	ADJ
ejpam-546	73	55	number	number	NOUN
ejpam-546	73	56	.	.	PUNCT
ejpam-546	74	1	the	the	DET
ejpam-546	74	2	general	general	ADJ
ejpam-546	74	3	pattern	pattern	NOUN
ejpam-546	74	4	n2	n2	ADJ
ejpam-546	74	5	+	+	CCONJ
ejpam-546	74	6	(	(	PUNCT
ejpam-546	74	7	n2	n2	ADJ
ejpam-546	74	8	+	+	X
ejpam-546	74	9	1	1	NUM
ejpam-546	74	10	)	)	PUNCT
ejpam-546	74	11	+	+	CCONJ
ejpam-546	74	12	.	.	PUNCT
ejpam-546	74	13	.	.	PUNCT
ejpam-546	75	1	.+	.+	NOUN
ejpam-546	75	2	(	(	PUNCT
ejpam-546	75	3	n2	n2	ADJ
ejpam-546	75	4	+	+	NUM
ejpam-546	75	5	n	n	CCONJ
ejpam-546	75	6	)	)	PUNCT
ejpam-546	75	7	=	=	SYM
ejpam-546	75	8	(	(	PUNCT
ejpam-546	75	9	n2	n2	PROPN
ejpam-546	75	10	+	+	X
ejpam-546	75	11	n+	n+	NUM
ejpam-546	75	12	1)+	1)+	NUM
ejpam-546	75	13	.	.	PUNCT
ejpam-546	75	14	.	.	PUNCT
ejpam-546	76	1	.+	.+	NOUN
ejpam-546	76	2	(	(	PUNCT
ejpam-546	76	3	n2	n2	ADJ
ejpam-546	76	4	+	+	X
ejpam-546	76	5	2n	2n	NUM
ejpam-546	76	6	)	)	PUNCT
ejpam-546	76	7	can	can	AUX
ejpam-546	76	8	be	be	AUX
ejpam-546	76	9	proved	prove	VERB
ejpam-546	76	10	by	by	ADP
ejpam-546	76	11	induction	induction	NOUN
ejpam-546	76	12	,	,	PUNCT
ejpam-546	76	13	but	but	CCONJ
ejpam-546	76	14	the	the	DET
ejpam-546	76	15	following	follow	VERB
ejpam-546	76	16	visual	visual	ADJ
ejpam-546	76	17	proof	proof	NOUN
ejpam-546	76	18	is	be	AUX
ejpam-546	76	19	much	much	ADV
ejpam-546	76	20	nicer	nice	ADJ
ejpam-546	76	21	.	.	PUNCT
ejpam-546	77	1	figure	figure	VERB
ejpam-546	77	2	4	4	NUM
ejpam-546	77	3	c.	c.	PROPN
ejpam-546	77	4	alsina	alsina	PROPN
ejpam-546	77	5	and	and	CCONJ
ejpam-546	77	6	r.	r.	PROPN
ejpam-546	77	7	nelsen	nelsen	PROPN
ejpam-546	77	8	/	/	SYM
ejpam-546	77	9	eur	eur	PROPN
ejpam-546	77	10	.	.	PUNCT
ejpam-546	78	1	j.	j.	PROPN
ejpam-546	78	2	pure	pure	PROPN
ejpam-546	78	3	appl	appl	PROPN
ejpam-546	78	4	.	.	PROPN
ejpam-546	78	5	math	math	PROPN
ejpam-546	78	6	,	,	PUNCT
ejpam-546	78	7	3	3	NUM
ejpam-546	78	8	(	(	PUNCT
ejpam-546	78	9	2010	2010	NUM
ejpam-546	78	10	)	)	PUNCT
ejpam-546	78	11	,	,	PUNCT
ejpam-546	78	12	118	118	NUM
ejpam-546	78	13	-	-	SYM
ejpam-546	78	14	127	127	NUM
ejpam-546	78	15	122	122	NUM
ejpam-546	78	16	in	in	ADP
ejpam-546	78	17	figure	figure	NOUN
ejpam-546	78	18	4	4	NUM
ejpam-546	78	19	,	,	PUNCT
ejpam-546	78	20	we	we	PRON
ejpam-546	78	21	see	see	VERB
ejpam-546	78	22	the	the	DET
ejpam-546	78	23	n	n	NOUN
ejpam-546	78	24	=	=	SYM
ejpam-546	78	25	4	4	NUM
ejpam-546	78	26	version	version	NOUN
ejpam-546	78	27	of	of	ADP
ejpam-546	78	28	the	the	DET
ejpam-546	78	29	identity	identity	NOUN
ejpam-546	78	30	where	where	SCONJ
ejpam-546	78	31	counting	count	VERB
ejpam-546	78	32	the	the	DET
ejpam-546	78	33	number	number	NOUN
ejpam-546	78	34	of	of	ADP
ejpam-546	78	35	small	small	ADJ
ejpam-546	78	36	cubes	cube	NOUN
ejpam-546	78	37	in	in	ADP
ejpam-546	78	38	the	the	DET
ejpam-546	78	39	pile	pile	NOUN
ejpam-546	78	40	in	in	ADP
ejpam-546	78	41	two	two	NUM
ejpam-546	78	42	different	different	ADJ
ejpam-546	78	43	ways	way	NOUN
ejpam-546	78	44	yields	yield	VERB
ejpam-546	78	45	16	16	NUM
ejpam-546	79	1	+	+	NUM
ejpam-546	79	2	17	17	NUM
ejpam-546	79	3	+	+	SYM
ejpam-546	79	4	18	18	NUM
ejpam-546	79	5	+	+	NUM
ejpam-546	79	6	19	19	NUM
ejpam-546	79	7	+	+	NOUN
ejpam-546	79	8	20=	20=	NUM
ejpam-546	79	9	21	21	NUM
ejpam-546	79	10	+	+	NUM
ejpam-546	79	11	22	22	NUM
ejpam-546	79	12	+	+	NUM
ejpam-546	79	13	23	23	NUM
ejpam-546	80	1	+	+	SYM
ejpam-546	80	2	24	24	NUM
ejpam-546	80	3	.	.	PUNCT
ejpam-546	81	1	there	there	PRON
ejpam-546	81	2	are	be	VERB
ejpam-546	81	3	many	many	ADJ
ejpam-546	81	4	nice	nice	ADJ
ejpam-546	81	5	relationships	relationship	NOUN
ejpam-546	81	6	between	between	ADP
ejpam-546	81	7	triangular	triangular	NOUN
ejpam-546	81	8	and	and	CCONJ
ejpam-546	81	9	square	square	ADJ
ejpam-546	81	10	numbers	number	NOUN
ejpam-546	81	11	.	.	PUNCT
ejpam-546	82	1	the	the	DET
ejpam-546	82	2	simplest	simplest	NOUN
ejpam-546	82	3	is	be	AUX
ejpam-546	82	4	perhaps	perhaps	ADV
ejpam-546	82	5	the	the	DET
ejpam-546	82	6	one	one	NOUN
ejpam-546	82	7	illustrated	illustrate	VERB
ejpam-546	82	8	in	in	ADP
ejpam-546	82	9	the	the	DET
ejpam-546	82	10	right	right	ADJ
ejpam-546	82	11	side	side	NOUN
ejpam-546	82	12	of	of	ADP
ejpam-546	82	13	figure	figure	NOUN
ejpam-546	82	14	3b	3b	NUM
ejpam-546	82	15	:	:	PUNCT
ejpam-546	82	16	tn−1	tn−1	PROPN
ejpam-546	82	17	+	+	CCONJ
ejpam-546	82	18	tn	tn	NOUN
ejpam-546	82	19	=	=	SYM
ejpam-546	82	20	n2	n2	NOUN
ejpam-546	82	21	.	.	PUNCT
ejpam-546	83	1	two	two	NUM
ejpam-546	83	2	more	more	ADJ
ejpam-546	83	3	are	be	AUX
ejpam-546	83	4	given	give	VERB
ejpam-546	83	5	in	in	ADP
ejpam-546	83	6	the	the	DET
ejpam-546	83	7	following	follow	VERB
ejpam-546	83	8	lemma	lemma	PROPN
ejpam-546	83	9	(	(	PUNCT
ejpam-546	83	10	setting	set	VERB
ejpam-546	83	11	t0	t0	PROPN
ejpam-546	83	12	=	=	SYM
ejpam-546	83	13	0	0	NUM
ejpam-546	83	14	for	for	ADP
ejpam-546	83	15	convenience	convenience	NOUN
ejpam-546	83	16	):	):	PUNCT
ejpam-546	83	17	lemma	lemma	PROPN
ejpam-546	83	18	1	1	NUM
ejpam-546	83	19	.	.	PUNCT
ejpam-546	83	20	for	for	ADP
ejpam-546	83	21	all	all	DET
ejpam-546	83	22	n≥	n≥	NOUN
ejpam-546	83	23	0	0	NUM
ejpam-546	83	24	,	,	PUNCT
ejpam-546	83	25	(	(	PUNCT
ejpam-546	83	26	a	a	PRON
ejpam-546	83	27	)	)	PUNCT
ejpam-546	83	28	8tn	8tn	NOUN
ejpam-546	83	29	=	=	SYM
ejpam-546	83	30	1=	1=	X
ejpam-546	83	31	(	(	PUNCT
ejpam-546	83	32	2n+	2n+	NUM
ejpam-546	83	33	1)2	1)2	NUM
ejpam-546	83	34	,	,	PUNCT
ejpam-546	83	35	and	and	CCONJ
ejpam-546	83	36	(	(	PUNCT
ejpam-546	83	37	b	b	X
ejpam-546	83	38	)	)	PUNCT
ejpam-546	83	39	9tn+	9tn+	PROPN
ejpam-546	83	40	1=	1=	X
ejpam-546	83	41	t3n+1	t3n+1	PROPN
ejpam-546	83	42	.	.	PUNCT
ejpam-546	84	1	proof	proof	NOUN
ejpam-546	84	2	.	.	PUNCT
ejpam-546	85	1	see	see	VERB
ejpam-546	85	2	figure	figure	NOUN
ejpam-546	85	3	5	5	NUM
ejpam-546	85	4	(	(	PUNCT
ejpam-546	85	5	where	where	SCONJ
ejpam-546	85	6	we	we	PRON
ejpam-546	85	7	have	have	AUX
ejpam-546	85	8	replaced	replace	VERB
ejpam-546	85	9	balls	ball	NOUN
ejpam-546	85	10	by	by	ADP
ejpam-546	85	11	squares	square	NOUN
ejpam-546	85	12	)	)	PUNCT
ejpam-546	85	13	.	.	PUNCT
ejpam-546	86	1	(	(	PUNCT
ejpam-546	86	2	a	a	X
ejpam-546	86	3	)	)	PUNCT
ejpam-546	86	4	(	(	PUNCT
ejpam-546	86	5	b	b	X
ejpam-546	86	6	)	)	PUNCT
ejpam-546	86	7	figure	figure	NOUN
ejpam-546	86	8	5	5	NUM
ejpam-546	86	9	lemma	lemma	PROPN
ejpam-546	86	10	1	1	NUM
ejpam-546	86	11	enables	enable	VERB
ejpam-546	86	12	us	we	PRON
ejpam-546	86	13	to	to	PART
ejpam-546	86	14	prove	prove	VERB
ejpam-546	86	15	the	the	DET
ejpam-546	86	16	following	follow	VERB
ejpam-546	86	17	two	two	NUM
ejpam-546	86	18	theorems	theorem	NOUN
ejpam-546	86	19	.	.	PUNCT
ejpam-546	87	1	theorem	theorem	NOUN
ejpam-546	87	2	3	3	X
ejpam-546	87	3	.	.	X
ejpam-546	88	1	there	there	PRON
ejpam-546	88	2	are	be	VERB
ejpam-546	88	3	infinitely	infinitely	ADV
ejpam-546	88	4	many	many	ADJ
ejpam-546	88	5	numbers	number	NOUN
ejpam-546	88	6	that	that	PRON
ejpam-546	88	7	are	be	AUX
ejpam-546	88	8	simultaneously	simultaneously	ADV
ejpam-546	88	9	square	square	ADJ
ejpam-546	88	10	and	and	CCONJ
ejpam-546	88	11	triangular	triangular	NOUN
ejpam-546	88	12	.	.	PUNCT
ejpam-546	89	1	proof	proof	NOUN
ejpam-546	89	2	.	.	PUNCT
ejpam-546	90	1	observe	observe	VERB
ejpam-546	90	2	that	that	SCONJ
ejpam-546	90	3	t8tn	t8tn	ADV
ejpam-546	91	1	=	=	SYM
ejpam-546	91	2	8tn(8tn+	8tn(8tn+	NUM
ejpam-546	91	3	1	1	NUM
ejpam-546	91	4	)	)	PUNCT
ejpam-546	91	5	2	2	NUM
ejpam-546	91	6	=	=	SYM
ejpam-546	91	7	4tn(2n+	4tn(2n+	NUM
ejpam-546	92	1	1)2	1)2	NUM
ejpam-546	92	2	,	,	PUNCT
ejpam-546	92	3	so	so	CCONJ
ejpam-546	92	4	if	if	SCONJ
ejpam-546	92	5	tn	tn	NOUN
ejpam-546	92	6	is	be	AUX
ejpam-546	92	7	square	square	ADJ
ejpam-546	92	8	,	,	PUNCT
ejpam-546	92	9	then	then	ADV
ejpam-546	92	10	so	so	ADV
ejpam-546	92	11	is	be	AUX
ejpam-546	92	12	t8tn	t8tn	ADV
ejpam-546	92	13	.	.	PUNCT
ejpam-546	93	1	since	since	SCONJ
ejpam-546	93	2	t1	t1	NOUN
ejpam-546	93	3	=	=	SYM
ejpam-546	93	4	1	1	NUM
ejpam-546	93	5	,	,	PUNCT
ejpam-546	93	6	this	this	DET
ejpam-546	93	7	relation	relation	NOUN
ejpam-546	93	8	generates	generate	VERB
ejpam-546	93	9	an	an	DET
ejpam-546	93	10	infinite	infinite	ADJ
ejpam-546	93	11	sequence	sequence	NOUN
ejpam-546	93	12	of	of	ADP
ejpam-546	93	13	square	square	ADJ
ejpam-546	93	14	triangular	triangular	NOUN
ejpam-546	93	15	numbers	number	NOUN
ejpam-546	93	16	,	,	PUNCT
ejpam-546	93	17	e.g.	e.g.	ADV
ejpam-546	93	18	,	,	PUNCT
ejpam-546	93	19	t8	t8	NOUN
ejpam-546	93	20	=	=	SYM
ejpam-546	93	21	62	62	NUM
ejpam-546	93	22	and	and	CCONJ
ejpam-546	93	23	t288	t288	PROPN
ejpam-546	93	24	=	=	SYM
ejpam-546	93	25	2042	2042	NUM
ejpam-546	93	26	.	.	PUNCT
ejpam-546	94	1	however	however	ADV
ejpam-546	94	2	,	,	PUNCT
ejpam-546	94	3	there	there	PRON
ejpam-546	94	4	are	be	VERB
ejpam-546	94	5	square	square	ADJ
ejpam-546	94	6	triangular	triangular	NOUN
ejpam-546	94	7	numbers	number	NOUN
ejpam-546	94	8	such	such	ADJ
ejpam-546	94	9	as	as	ADP
ejpam-546	94	10	t49	t49	PROPN
ejpam-546	94	11	=	=	SYM
ejpam-546	94	12	352	352	NUM
ejpam-546	95	1	and	and	CCONJ
ejpam-546	95	2	t1681	t1681	NOUN
ejpam-546	95	3	=	=	SYM
ejpam-546	95	4	11892	11892	NUM
ejpam-546	95	5	that	that	PRON
ejpam-546	95	6	are	be	AUX
ejpam-546	95	7	not	not	PART
ejpam-546	95	8	in	in	ADP
ejpam-546	95	9	this	this	DET
ejpam-546	95	10	sequence	sequence	NOUN
ejpam-546	95	11	.	.	PUNCT
ejpam-546	96	1	theorem	theorem	ADJ
ejpam-546	96	2	4	4	NUM
ejpam-546	96	3	.	.	PUNCT
ejpam-546	96	4	sums	sum	NOUN
ejpam-546	96	5	of	of	ADP
ejpam-546	96	6	powers	power	NOUN
ejpam-546	96	7	of	of	ADP
ejpam-546	96	8	9	9	NUM
ejpam-546	96	9	are	be	AUX
ejpam-546	96	10	triangular	triangular	NOUN
ejpam-546	96	11	numbers	number	NOUN
ejpam-546	96	12	,	,	PUNCT
ejpam-546	96	13	i.e.	i.e.	X
ejpam-546	96	14	,	,	PUNCT
ejpam-546	96	15	for	for	ADP
ejpam-546	96	16	all	all	DET
ejpam-546	96	17	n≥	n≥	NOUN
ejpam-546	96	18	0	0	NUM
ejpam-546	96	19	,	,	PUNCT
ejpam-546	96	20	1	1	NUM
ejpam-546	96	21	+	+	NUM
ejpam-546	96	22	9	9	NUM
ejpam-546	96	23	+	+	NUM
ejpam-546	96	24	92	92	NUM
ejpam-546	96	25	+	+	NUM
ejpam-546	96	26	.	.	PUNCT
ejpam-546	96	27	.	.	PUNCT
ejpam-546	97	1	.+	.+	NOUN
ejpam-546	97	2	9n	9n	NUM
ejpam-546	97	3	=	=	PUNCT
ejpam-546	98	1	t1	t1	NUM
ejpam-546	98	2	+	+	PROPN
ejpam-546	98	3	3	3	NUM
ejpam-546	98	4	+	+	NOUN
ejpam-546	98	5	32+	32+	NUM
ejpam-546	98	6	...	...	PUNCT
ejpam-546	98	7	+3n	+3n	NUM
ejpam-546	98	8	proof	proof	NOUN
ejpam-546	98	9	.	.	PUNCT
ejpam-546	99	1	see	see	VERB
ejpam-546	99	2	figure	figure	NOUN
ejpam-546	99	3	6	6	NUM
ejpam-546	99	4	.	.	PUNCT
ejpam-546	100	1	figure	figure	VERB
ejpam-546	100	2	6	6	NUM
ejpam-546	100	3	c.	c.	PROPN
ejpam-546	100	4	alsina	alsina	PROPN
ejpam-546	100	5	and	and	CCONJ
ejpam-546	100	6	r.	r.	PROPN
ejpam-546	100	7	nelsen	nelsen	PROPN
ejpam-546	100	8	/	/	SYM
ejpam-546	100	9	eur	eur	PROPN
ejpam-546	100	10	.	.	PUNCT
ejpam-546	101	1	j.	j.	PROPN
ejpam-546	101	2	pure	pure	PROPN
ejpam-546	101	3	appl	appl	PROPN
ejpam-546	101	4	.	.	PROPN
ejpam-546	101	5	math	math	PROPN
ejpam-546	101	6	,	,	PUNCT
ejpam-546	101	7	3	3	NUM
ejpam-546	101	8	(	(	PUNCT
ejpam-546	101	9	2010	2010	NUM
ejpam-546	101	10	)	)	PUNCT
ejpam-546	101	11	,	,	PUNCT
ejpam-546	101	12	118	118	NUM
ejpam-546	101	13	-	-	SYM
ejpam-546	101	14	127	127	NUM
ejpam-546	101	15	123	123	NUM
ejpam-546	101	16	as	as	ADP
ejpam-546	101	17	a	a	DET
ejpam-546	101	18	consequence	consequence	NOUN
ejpam-546	101	19	,	,	PUNCT
ejpam-546	101	20	in	in	ADP
ejpam-546	101	21	base	base	NOUN
ejpam-546	101	22	9	9	NUM
ejpam-546	101	23	the	the	DET
ejpam-546	101	24	numbers	number	NOUN
ejpam-546	101	25	1,11,111,1111	1,11,111,1111	NUM
ejpam-546	101	26	,	,	PUNCT
ejpam-546	101	27	.	.	PUNCT
ejpam-546	101	28	.	.	PUNCT
ejpam-546	102	1	.	.	PUNCT
ejpam-546	103	1	are	be	AUX
ejpam-546	103	2	all	all	ADV
ejpam-546	103	3	triangular	triangular	NOUN
ejpam-546	103	4	.	.	PUNCT
ejpam-546	104	1	the	the	DET
ejpam-546	104	2	next	next	ADJ
ejpam-546	104	3	theorem	theorem	NOUN
ejpam-546	104	4	presents	present	VERB
ejpam-546	104	5	a	a	DET
ejpam-546	104	6	companion	companion	NOUN
ejpam-546	104	7	to	to	ADP
ejpam-546	104	8	the	the	DET
ejpam-546	104	9	identity	identity	NOUN
ejpam-546	104	10	tn−1	tn−1	PROPN
ejpam-546	104	11	+	+	CCONJ
ejpam-546	104	12	tn	tn	NOUN
ejpam-546	104	13	=	=	SYM
ejpam-546	104	14	n2	n2	PROPN
ejpam-546	104	15	.	.	PUNCT
ejpam-546	105	1	theorem	theorem	NOUN
ejpam-546	105	2	5	5	NUM
ejpam-546	105	3	.	.	PUNCT
ejpam-546	106	1	the	the	DET
ejpam-546	106	2	sum	sum	NOUN
ejpam-546	106	3	of	of	ADP
ejpam-546	106	4	the	the	DET
ejpam-546	106	5	squares	square	NOUN
ejpam-546	106	6	of	of	ADP
ejpam-546	106	7	consecutive	consecutive	ADJ
ejpam-546	106	8	triangular	triangular	NOUN
ejpam-546	106	9	numbers	number	NOUN
ejpam-546	106	10	is	be	AUX
ejpam-546	106	11	a	a	DET
ejpam-546	106	12	triangular	triangular	ADJ
ejpam-546	106	13	number	number	NOUN
ejpam-546	106	14	,	,	PUNCT
ejpam-546	106	15	i.e.	i.e.	X
ejpam-546	106	16	,	,	PUNCT
ejpam-546	106	17	t2	t2	PROPN
ejpam-546	106	18	n−1	n−1	PROPN
ejpam-546	106	19	+	+	CCONJ
ejpam-546	106	20	t2	t2	PROPN
ejpam-546	106	21	n	n	NOUN
ejpam-546	106	22	=	=	SYM
ejpam-546	106	23	tn2	tn2	ADJ
ejpam-546	106	24	for	for	ADP
ejpam-546	106	25	all	all	PRON
ejpam-546	106	26	n≥	n≥	ADJ
ejpam-546	106	27	1	1	NUM
ejpam-546	106	28	.	.	PUNCT
ejpam-546	107	1	proof	proof	NOUN
ejpam-546	107	2	.	.	PUNCT
ejpam-546	108	1	see	see	VERB
ejpam-546	108	2	figure	figure	NOUN
ejpam-546	108	3	7	7	NUM
ejpam-546	108	4	,	,	PUNCT
ejpam-546	108	5	where	where	SCONJ
ejpam-546	108	6	we	we	PRON
ejpam-546	108	7	illustrate	illustrate	VERB
ejpam-546	108	8	the	the	DET
ejpam-546	108	9	square	square	NOUN
ejpam-546	108	10	of	of	ADP
ejpam-546	108	11	a	a	DET
ejpam-546	108	12	triangular	triangular	NOUN
ejpam-546	108	13	number	number	NOUN
ejpam-546	108	14	as	as	ADP
ejpam-546	108	15	a	a	DET
ejpam-546	108	16	triangular	triangular	NOUN
ejpam-546	108	17	array	array	NOUN
ejpam-546	108	18	of	of	ADP
ejpam-546	108	19	triangular	triangular	NOUN
ejpam-546	108	20	numbers	number	NOUN
ejpam-546	108	21	.	.	PUNCT
ejpam-546	109	1	figure	figure	VERB
ejpam-546	109	2	7	7	NUM
ejpam-546	109	3	you	you	PRON
ejpam-546	109	4	may	may	AUX
ejpam-546	109	5	have	have	AUX
ejpam-546	109	6	noticed	notice	VERB
ejpam-546	109	7	that	that	SCONJ
ejpam-546	109	8	the	the	DET
ejpam-546	109	9	nth	nth	NOUN
ejpam-546	109	10	triangular	triangular	NOUN
ejpam-546	109	11	number	number	NOUN
ejpam-546	109	12	is	be	AUX
ejpam-546	109	13	a	a	DET
ejpam-546	109	14	binomial	binomial	ADJ
ejpam-546	109	15	coefficient	coefficient	NOUN
ejpam-546	109	16	,	,	PUNCT
ejpam-546	109	17	i.e.	i.e.	X
ejpam-546	109	18	,	,	PUNCT
ejpam-546	109	19	tn	tn	NOUN
ejpam-546	109	20	=	=	SYM
ejpam-546	109	21	�	�	PROPN
ejpam-546	109	22	n+1	n+1	PROPN
ejpam-546	109	23	2	2	NUM
ejpam-546	109	24	�	�	PROPN
ejpam-546	109	25	.	.	PUNCT
ejpam-546	110	1	one	one	NUM
ejpam-546	110	2	explanation	explanation	NOUN
ejpam-546	110	3	for	for	ADP
ejpam-546	110	4	this	this	PRON
ejpam-546	110	5	is	be	AUX
ejpam-546	110	6	that	that	SCONJ
ejpam-546	110	7	each	each	PRON
ejpam-546	110	8	is	be	AUX
ejpam-546	110	9	equal	equal	ADJ
ejpam-546	110	10	to	to	ADP
ejpam-546	110	11	n(n+1)/2	n(n+1)/2	NOUN
ejpam-546	110	12	,	,	PUNCT
ejpam-546	110	13	but	but	CCONJ
ejpam-546	110	14	this	this	DET
ejpam-546	110	15	answer	answer	NOUN
ejpam-546	110	16	sheds	shed	VERB
ejpam-546	110	17	little	little	ADJ
ejpam-546	110	18	light	light	NOUN
ejpam-546	110	19	on	on	ADP
ejpam-546	110	20	why	why	SCONJ
ejpam-546	110	21	it	it	PRON
ejpam-546	110	22	is	be	AUX
ejpam-546	110	23	true	true	ADJ
ejpam-546	110	24	.	.	PUNCT
ejpam-546	111	1	here	here	ADV
ejpam-546	111	2	is	be	AUX
ejpam-546	111	3	a	a	DET
ejpam-546	111	4	better	well	ADJ
ejpam-546	111	5	explanation	explanation	NOUN
ejpam-546	111	6	using	use	VERB
ejpam-546	111	7	the	the	DET
ejpam-546	111	8	cantor	cantor	PROPN
ejpam-546	111	9	principle	principle	NOUN
ejpam-546	111	10	:	:	PUNCT
ejpam-546	111	11	theorem	theorem	NOUN
ejpam-546	111	12	6	6	NUM
ejpam-546	111	13	.	.	PUNCT
ejpam-546	112	1	there	there	PRON
ejpam-546	112	2	exists	exist	VERB
ejpam-546	112	3	a	a	DET
ejpam-546	112	4	one	one	NUM
ejpam-546	112	5	-	-	PUNCT
ejpam-546	112	6	to	to	ADP
ejpam-546	112	7	-	-	PUNCT
ejpam-546	112	8	one	one	NUM
ejpam-546	112	9	correspondence	correspondence	NOUN
ejpam-546	112	10	between	between	ADP
ejpam-546	112	11	a	a	DET
ejpam-546	112	12	set	set	NOUN
ejpam-546	112	13	of	of	ADP
ejpam-546	112	14	tn	tn	NOUN
ejpam-546	112	15	objects	object	NOUN
ejpam-546	112	16	and	and	CCONJ
ejpam-546	112	17	the	the	DET
ejpam-546	112	18	set	set	NOUN
ejpam-546	112	19	of	of	ADP
ejpam-546	112	20	two	two	NUM
ejpam-546	112	21	-	-	PUNCT
ejpam-546	112	22	element	element	NOUN
ejpam-546	112	23	subsets	subset	NOUN
ejpam-546	112	24	of	of	ADP
ejpam-546	112	25	a	a	DET
ejpam-546	112	26	set	set	NOUN
ejpam-546	112	27	with	with	ADP
ejpam-546	112	28	n+	n+	ADP
ejpam-546	112	29	1	1	NUM
ejpam-546	112	30	objects	object	NOUN
ejpam-546	112	31	.	.	PUNCT
ejpam-546	113	1	proof	proof	NOUN
ejpam-546	113	2	.	.	PUNCT
ejpam-546	114	1	see	see	VERB
ejpam-546	114	2	figure	figure	NOUN
ejpam-546	114	3	8	8	NUM
ejpam-546	115	1	[	[	X
ejpam-546	115	2	6	6	NUM
ejpam-546	115	3	]	]	PUNCT
ejpam-546	115	4	,	,	PUNCT
ejpam-546	115	5	and	and	CCONJ
ejpam-546	115	6	recall	recall	VERB
ejpam-546	115	7	that	that	SCONJ
ejpam-546	115	8	the	the	DET
ejpam-546	115	9	binomial	binomial	ADJ
ejpam-546	115	10	coefficient	coefficient	NOUN
ejpam-546	115	11	�	�	PROPN
ejpam-546	115	12	k	k	PROPN
ejpam-546	115	13	2	2	NUM
ejpam-546	115	14	�	�	NOUN
ejpam-546	115	15	is	be	AUX
ejpam-546	115	16	the	the	DET
ejpam-546	115	17	number	number	NOUN
ejpam-546	115	18	of	of	ADP
ejpam-546	115	19	ways	way	NOUN
ejpam-546	115	20	to	to	PART
ejpam-546	115	21	choose	choose	VERB
ejpam-546	115	22	2	2	NUM
ejpam-546	115	23	elements	element	NOUN
ejpam-546	115	24	from	from	ADP
ejpam-546	115	25	a	a	DET
ejpam-546	115	26	set	set	NOUN
ejpam-546	115	27	of	of	ADP
ejpam-546	115	28	k	k	PROPN
ejpam-546	115	29	elements	element	NOUN
ejpam-546	115	30	.	.	PUNCT
ejpam-546	116	1	the	the	DET
ejpam-546	116	2	arrows	arrow	NOUN
ejpam-546	116	3	denote	denote	VERB
ejpam-546	116	4	the	the	DET
ejpam-546	116	5	correspondence	correspondence	NOUN
ejpam-546	116	6	between	between	ADP
ejpam-546	116	7	an	an	DET
ejpam-546	116	8	element	element	NOUN
ejpam-546	116	9	of	of	ADP
ejpam-546	116	10	the	the	DET
ejpam-546	116	11	set	set	NOUN
ejpam-546	116	12	with	with	ADP
ejpam-546	116	13	tn	tn	NOUN
ejpam-546	116	14	elements	element	NOUN
ejpam-546	116	15	and	and	CCONJ
ejpam-546	116	16	a	a	DET
ejpam-546	116	17	pair	pair	NOUN
ejpam-546	116	18	of	of	ADP
ejpam-546	116	19	elements	element	NOUN
ejpam-546	116	20	from	from	ADP
ejpam-546	116	21	a	a	DET
ejpam-546	116	22	set	set	NOUN
ejpam-546	116	23	of	of	ADP
ejpam-546	116	24	n+	n+	PRON
ejpam-546	116	25	1	1	NUM
ejpam-546	116	26	elements	element	NOUN
ejpam-546	116	27	.	.	PUNCT
ejpam-546	117	1	1	1	NUM
ejpam-546	117	2	2	2	NUM
ejpam-546	117	3	3	3	NUM
ejpam-546	117	4	n	n	SYM
ejpam-546	117	5	n+1	n+1	PROPN
ejpam-546	117	6	..	..	PUNCT
ejpam-546	117	7	.	.	PUNCT
ejpam-546	118	1	figure	figure	VERB
ejpam-546	118	2	8	8	NUM
ejpam-546	118	3	4	4	NUM
ejpam-546	118	4	.	.	PUNCT
ejpam-546	119	1	sums	sum	NOUN
ejpam-546	119	2	of	of	ADP
ejpam-546	119	3	squares	square	NOUN
ejpam-546	119	4	,	,	PUNCT
ejpam-546	119	5	triangular	triangular	NOUN
ejpam-546	119	6	numbers	number	NOUN
ejpam-546	119	7	,	,	PUNCT
ejpam-546	119	8	and	and	CCONJ
ejpam-546	119	9	cubes	cube	NOUN
ejpam-546	119	10	having	having	AUX
ejpam-546	119	11	examined	examine	VERB
ejpam-546	119	12	triangular	triangular	NOUN
ejpam-546	119	13	numbers	number	NOUN
ejpam-546	119	14	and	and	CCONJ
ejpam-546	119	15	squares	square	NOUN
ejpam-546	119	16	as	as	ADP
ejpam-546	119	17	sums	sum	NOUN
ejpam-546	119	18	of	of	ADP
ejpam-546	119	19	integers	integer	NOUN
ejpam-546	119	20	and	and	CCONJ
ejpam-546	119	21	sums	sum	NOUN
ejpam-546	119	22	of	of	ADP
ejpam-546	119	23	odd	odd	ADJ
ejpam-546	119	24	integers	integer	NOUN
ejpam-546	119	25	,	,	PUNCT
ejpam-546	119	26	we	we	PRON
ejpam-546	119	27	now	now	ADV
ejpam-546	119	28	consider	consider	VERB
ejpam-546	119	29	sums	sum	NOUN
ejpam-546	119	30	of	of	ADP
ejpam-546	119	31	triangular	triangular	NOUN
ejpam-546	119	32	numbers	number	NOUN
ejpam-546	119	33	and	and	CCONJ
ejpam-546	119	34	sums	sum	NOUN
ejpam-546	119	35	of	of	ADP
ejpam-546	119	36	squares	square	NOUN
ejpam-546	119	37	.	.	PUNCT
ejpam-546	120	1	c.	c.	PROPN
ejpam-546	120	2	alsina	alsina	PROPN
ejpam-546	120	3	and	and	CCONJ
ejpam-546	120	4	r.	r.	PROPN
ejpam-546	120	5	nelsen	nelsen	PROPN
ejpam-546	120	6	/	/	SYM
ejpam-546	120	7	eur	eur	PROPN
ejpam-546	120	8	.	.	PUNCT
ejpam-546	121	1	j.	j.	PROPN
ejpam-546	121	2	pure	pure	PROPN
ejpam-546	121	3	appl	appl	PROPN
ejpam-546	121	4	.	.	PROPN
ejpam-546	121	5	math	math	PROPN
ejpam-546	121	6	,	,	PUNCT
ejpam-546	121	7	3	3	NUM
ejpam-546	121	8	(	(	PUNCT
ejpam-546	121	9	2010	2010	NUM
ejpam-546	121	10	)	)	PUNCT
ejpam-546	121	11	,	,	PUNCT
ejpam-546	121	12	118	118	NUM
ejpam-546	121	13	-	-	SYM
ejpam-546	121	14	127	127	NUM
ejpam-546	121	15	124	124	NUM
ejpam-546	121	16	theorem	theorem	NOUN
ejpam-546	121	17	7	7	NUM
ejpam-546	121	18	.	.	PUNCT
ejpam-546	121	19	for	for	ADP
ejpam-546	121	20	all	all	DET
ejpam-546	121	21	n≥	n≥	ADJ
ejpam-546	121	22	1	1	NUM
ejpam-546	121	23	,	,	PUNCT
ejpam-546	121	24	12	12	NUM
ejpam-546	121	25	+	+	SYM
ejpam-546	121	26	22	22	NUM
ejpam-546	121	27	+	+	NUM
ejpam-546	121	28	32	32	NUM
ejpam-546	121	29	+	+	CCONJ
ejpam-546	121	30	.	.	PUNCT
ejpam-546	121	31	.	.	PUNCT
ejpam-546	122	1	.+	.+	NOUN
ejpam-546	122	2	n2	n2	PROPN
ejpam-546	122	3	=	=	PUNCT
ejpam-546	122	4	n(n+	n(n+	PROPN
ejpam-546	122	5	1)(2n+	1)(2n+	NUM
ejpam-546	122	6	1)/6	1)/6	NUM
ejpam-546	122	7	.	.	PUNCT
ejpam-546	123	1	proof	proof	NOUN
ejpam-546	123	2	.	.	PUNCT
ejpam-546	124	1	we	we	PRON
ejpam-546	124	2	give	give	VERB
ejpam-546	124	3	two	two	NUM
ejpam-546	124	4	proofs	proof	NOUN
ejpam-546	124	5	.	.	PUNCT
ejpam-546	125	1	the	the	DET
ejpam-546	125	2	first	first	ADJ
ejpam-546	125	3	is	be	AUX
ejpam-546	125	4	in	in	ADP
ejpam-546	125	5	figure	figure	NOUN
ejpam-546	125	6	9	9	NUM
ejpam-546	125	7	.	.	PUNCT
ejpam-546	126	1	aa	aa	NOUN
ejpam-546	126	2	aa	aa	PROPN
ejpam-546	126	3	aa	aa	PROPN
ejpam-546	126	4	aaa	aaa	NOUN
ejpam-546	126	5	aaa	aaa	NOUN
ejpam-546	126	6	aaa	aaa	NOUN
ejpam-546	126	7	aa	aa	NOUN
ejpam-546	126	8	aa	aa	PROPN
ejpam-546	126	9	aaa	aaa	NOUN
ejpam-546	126	10	aaa	aaa	NOUN
ejpam-546	126	11	aaa	aaa	NOUN
ejpam-546	126	12	aaa	aaa	NOUN
ejpam-546	126	13	aaa	aaa	NOUN
ejpam-546	126	14	aaa	aaa	NOUN
ejpam-546	126	15	aaa	aaa	NOUN
ejpam-546	126	16	aaa	aaa	NOUN
ejpam-546	126	17	aaa	aaa	NOUN
ejpam-546	126	18	aaa	aaa	NOUN
ejpam-546	126	19	aaa	aaa	NOUN
ejpam-546	126	20	aaa	aaa	NOUN
ejpam-546	126	21	aaa	aaa	NOUN
ejpam-546	126	22	aa	aa	NOUN
ejpam-546	126	23	aa	aa	PROPN
ejpam-546	126	24	aaa	aaa	NOUN
ejpam-546	126	25	aaa	aaa	NOUN
ejpam-546	126	26	aaa	aaa	NOUN
ejpam-546	126	27	aaaaaa	aaaaaa	NOUN
ejpam-546	126	28	aaa	aaa	NOUN
ejpam-546	126	29	aaa	aaa	NOUN
ejpam-546	126	30	figure	figure	NOUN
ejpam-546	126	31	9	9	NUM
ejpam-546	126	32	we	we	PRON
ejpam-546	126	33	exhibit	exhibit	VERB
ejpam-546	126	34	a	a	DET
ejpam-546	126	35	one	one	NUM
ejpam-546	126	36	-	-	PUNCT
ejpam-546	126	37	to	to	ADP
ejpam-546	126	38	-	-	PUNCT
ejpam-546	126	39	one	one	NUM
ejpam-546	126	40	correspondence	correspondence	NOUN
ejpam-546	126	41	between	between	ADP
ejpam-546	126	42	three	three	NUM
ejpam-546	126	43	copies	copy	NOUN
ejpam-546	126	44	of	of	ADP
ejpam-546	126	45	12	12	NUM
ejpam-546	126	46	+	+	NUM
ejpam-546	126	47	22	22	NUM
ejpam-546	126	48	+	+	NUM
ejpam-546	126	49	32	32	NUM
ejpam-546	126	50	+	+	CCONJ
ejpam-546	126	51	.	.	PUNCT
ejpam-546	126	52	.	.	PUNCT
ejpam-546	126	53	.	.	PUNCT
ejpam-546	127	1	+	+	CCONJ
ejpam-546	127	2	n2	n2	ADJ
ejpam-546	127	3	and	and	CCONJ
ejpam-546	127	4	a	a	DET
ejpam-546	127	5	rectangle	rectangle	NOUN
ejpam-546	127	6	whose	whose	DET
ejpam-546	127	7	dimensions	dimension	NOUN
ejpam-546	127	8	are	be	AUX
ejpam-546	127	9	2n+	2n+	NUM
ejpam-546	127	10	1	1	NUM
ejpam-546	127	11	and	and	CCONJ
ejpam-546	127	12	1	1	NUM
ejpam-546	127	13	+	+	NUM
ejpam-546	127	14	2	2	NUM
ejpam-546	127	15	+	+	NUM
ejpam-546	127	16	.	.	PUNCT
ejpam-546	127	17	.	.	PUNCT
ejpam-546	127	18	.	.	PUNCT
ejpam-546	128	1	+	+	CCONJ
ejpam-546	128	2	n	n	CCONJ
ejpam-546	128	3	=	=	SYM
ejpam-546	128	4	n(n+	n(n+	X
ejpam-546	128	5	1)/2	1)/2	NUM
ejpam-546	128	6	[	[	X
ejpam-546	128	7	4	4	NUM
ejpam-546	128	8	]	]	PUNCT
ejpam-546	128	9	.	.	PUNCT
ejpam-546	129	1	hence	hence	ADV
ejpam-546	129	2	3(12	3(12	NUM
ejpam-546	129	3	+	+	NOUN
ejpam-546	129	4	22	22	NUM
ejpam-546	129	5	+	+	NUM
ejpam-546	129	6	32	32	NUM
ejpam-546	129	7	+	+	NUM
ejpam-546	129	8	.	.	PUNCT
ejpam-546	129	9	.	.	PUNCT
ejpam-546	130	1	.+	.+	NOUN
ejpam-546	130	2	n2	n2	NOUN
ejpam-546	130	3	)	)	PUNCT
ejpam-546	130	4	=	=	PUNCT
ejpam-546	130	5	(	(	PUNCT
ejpam-546	130	6	2n+	2n+	NUM
ejpam-546	130	7	1)(1	1)(1	NUM
ejpam-546	130	8	+	+	NOUN
ejpam-546	130	9	2	2	NUM
ejpam-546	130	10	+	+	NUM
ejpam-546	130	11	.	.	PUNCT
ejpam-546	130	12	.	.	PUNCT
ejpam-546	131	1	.+	.+	NOUN
ejpam-546	131	2	n	n	CCONJ
ejpam-546	131	3	)	)	PUNCT
ejpam-546	131	4	from	from	ADP
ejpam-546	131	5	which	which	PRON
ejpam-546	131	6	the	the	DET
ejpam-546	131	7	result	result	NOUN
ejpam-546	131	8	now	now	ADV
ejpam-546	131	9	follows	follow	VERB
ejpam-546	131	10	.	.	PUNCT
ejpam-546	132	1	for	for	ADP
ejpam-546	132	2	a	a	DET
ejpam-546	132	3	second	second	ADJ
ejpam-546	132	4	proof	proof	NOUN
ejpam-546	132	5	,	,	PUNCT
ejpam-546	132	6	see	see	VERB
ejpam-546	132	7	figure	figure	NOUN
ejpam-546	132	8	10	10	NUM
ejpam-546	133	1	[	[	X
ejpam-546	133	2	5	5	NUM
ejpam-546	133	3	]	]	PUNCT
ejpam-546	133	4	.	.	PUNCT
ejpam-546	134	1	2	2	NUM
ejpam-546	134	2	3	3	NUM
ejpam-546	134	3	4	4	NUM
ejpam-546	134	4	n-1	n-1	NOUN
ejpam-546	134	5	1	1	NUM
ejpam-546	134	6	2	2	NUM
ejpam-546	134	7	3	3	NUM
ejpam-546	134	8	n	n	NUM
ejpam-546	134	9	n	n	PROPN
ejpam-546	134	10	..	..	PUNCT
ejpam-546	134	11	.	.	PUNCT
ejpam-546	134	12	...	...	PUNCT
ejpam-546	134	13	...	...	PUNCT
ejpam-546	135	1	n-1	n-1	X
ejpam-546	136	1	n	n	PRON
ejpam-546	136	2	n-1	n-1	NOUN
ejpam-546	136	3	n	n	PRON
ejpam-546	136	4	n-2	n-2	X
ejpam-546	136	5	n-1	n-1	NOUN
ejpam-546	136	6	n+	n+	PUNCT
ejpam-546	136	7	2n+1	2n+1	PROPN
ejpam-546	136	8	2n+1	2n+1	PROPN
ejpam-546	136	9	2n+1	2n+1	PROPN
ejpam-546	136	10	..	..	PUNCT
ejpam-546	136	11	.	.	PUNCT
ejpam-546	136	12	...	...	PUNCT
ejpam-546	136	13	...	...	PUNCT
ejpam-546	137	1	2n+1	2n+1	PROPN
ejpam-546	137	2	2n+1	2n+1	PROPN
ejpam-546	137	3	2n+1	2n+1	PROPN
ejpam-546	137	4	2n+1	2n+1	PROPN
ejpam-546	137	5	2n+1	2n+1	PROPN
ejpam-546	137	6	2n+1	2n+1	PROPN
ejpam-546	137	7	2n+1	2n+1	PROPN
ejpam-546	137	8	2n+1	2n+1	PROPN
ejpam-546	137	9	2n+1	2n+1	PROPN
ejpam-546	138	1	=	=	PUNCT
ejpam-546	139	1	+	+	CCONJ
ejpam-546	139	2	1	1	NUM
ejpam-546	139	3	2	2	NUM
ejpam-546	139	4	2	2	NUM
ejpam-546	139	5	3	3	NUM
ejpam-546	139	6	3	3	NUM
ejpam-546	139	7	3	3	NUM
ejpam-546	139	8	n-1	n-1	NOUN
ejpam-546	139	9	n-1	n-1	VERB
ejpam-546	139	10	n	n	CCONJ
ejpam-546	139	11	n	n	CCONJ
ejpam-546	139	12	n	n	CCONJ
ejpam-546	139	13	n	n	CCONJ
ejpam-546	139	14	n	n	PROPN
ejpam-546	139	15	..	..	PUNCT
ejpam-546	139	16	.	.	PUNCT
ejpam-546	139	17	...	...	PUNCT
ejpam-546	140	1	...	...	PUNCT
ejpam-546	141	1	n-1	n-1	PROPN
ejpam-546	141	2	n	n	CCONJ
ejpam-546	141	3	n	n	PRON
ejpam-546	141	4	n-1	n-1	NOUN
ejpam-546	141	5	n	n	CCONJ
ejpam-546	141	6	n-2	n-2	PROPN
ejpam-546	141	7	234n	234n	NUM
ejpam-546	141	8	123n-1n	123n-1n	NUM
ejpam-546	141	9	..	..	PUNCT
ejpam-546	141	10	.	.	PUNCT
ejpam-546	141	11	...	...	PUNCT
ejpam-546	142	1	...	...	PUNCT
ejpam-546	142	2	n-1	n-1	X
ejpam-546	142	3	n-1	n-1	X
ejpam-546	142	4	figure	figure	VERB
ejpam-546	142	5	10	10	NUM
ejpam-546	142	6	here	here	ADV
ejpam-546	142	7	we	we	PRON
ejpam-546	142	8	write	write	VERB
ejpam-546	142	9	each	each	DET
ejpam-546	142	10	square	square	ADJ
ejpam-546	142	11	k2	k2	NOUN
ejpam-546	142	12	as	as	ADP
ejpam-546	142	13	a	a	DET
ejpam-546	142	14	sum	sum	NOUN
ejpam-546	142	15	of	of	ADP
ejpam-546	142	16	k	k	PROPN
ejpam-546	142	17	ks	ks	PROPN
ejpam-546	142	18	,	,	PUNCT
ejpam-546	142	19	then	then	ADV
ejpam-546	142	20	place	place	VERB
ejpam-546	142	21	those	those	DET
ejpam-546	142	22	numbers	number	NOUN
ejpam-546	142	23	in	in	ADP
ejpam-546	142	24	a	a	DET
ejpam-546	142	25	triangular	triangular	NOUN
ejpam-546	142	26	array	array	NOUN
ejpam-546	142	27	,	,	PUNCT
ejpam-546	142	28	create	create	VERB
ejpam-546	142	29	two	two	NUM
ejpam-546	142	30	more	more	ADJ
ejpam-546	142	31	arrays	array	NOUN
ejpam-546	142	32	by	by	ADP
ejpam-546	142	33	rotating	rotate	VERB
ejpam-546	142	34	the	the	DET
ejpam-546	142	35	triangular	triangular	NOUN
ejpam-546	142	36	array	array	NOUN
ejpam-546	142	37	by	by	ADP
ejpam-546	142	38	120	120	NUM
ejpam-546	142	39	◦	◦	NOUN
ejpam-546	142	40	and	and	CCONJ
ejpam-546	142	41	240	240	NUM
ejpam-546	142	42	◦	◦	NOUN
ejpam-546	142	43	,	,	PUNCT
ejpam-546	142	44	and	and	CCONJ
ejpam-546	142	45	add	add	VERB
ejpam-546	142	46	corresponding	corresponding	ADJ
ejpam-546	142	47	entries	entry	NOUN
ejpam-546	142	48	in	in	ADP
ejpam-546	142	49	each	each	DET
ejpam-546	142	50	triangular	triangular	NOUN
ejpam-546	142	51	array	array	NOUN
ejpam-546	142	52	.	.	PUNCT
ejpam-546	143	1	theorem	theorem	ADJ
ejpam-546	143	2	8	8	NUM
ejpam-546	143	3	.	.	PUNCT
ejpam-546	144	1	for	for	ADP
ejpam-546	144	2	all	all	DET
ejpam-546	144	3	n≥	n≥	ADJ
ejpam-546	144	4	1	1	NUM
ejpam-546	144	5	,	,	PUNCT
ejpam-546	144	6	t1	t1	NOUN
ejpam-546	144	7	+	+	CCONJ
ejpam-546	144	8	t2	t2	NOUN
ejpam-546	144	9	+	+	CCONJ
ejpam-546	144	10	t3	t3	PROPN
ejpam-546	144	11	+	+	X
ejpam-546	144	12	.	.	PUNCT
ejpam-546	144	13	.	.	PUNCT
ejpam-546	145	1	.+	.+	NOUN
ejpam-546	145	2	tn	tn	NOUN
ejpam-546	146	1	=	=	PUNCT
ejpam-546	146	2	n(n+	n(n+	PROPN
ejpam-546	146	3	1)(n+	1)(n+	NUM
ejpam-546	146	4	2)/6	2)/6	NUM
ejpam-546	146	5	.	.	PUNCT
ejpam-546	147	1	proof	proof	NOUN
ejpam-546	147	2	.	.	PUNCT
ejpam-546	148	1	see	see	VERB
ejpam-546	148	2	figure	figure	NOUN
ejpam-546	148	3	11	11	NUM
ejpam-546	148	4	.	.	PUNCT
ejpam-546	149	1	c.	c.	PROPN
ejpam-546	149	2	alsina	alsina	PROPN
ejpam-546	149	3	and	and	CCONJ
ejpam-546	149	4	r.	r.	PROPN
ejpam-546	149	5	nelsen	nelsen	PROPN
ejpam-546	149	6	/	/	SYM
ejpam-546	149	7	eur	eur	PROPN
ejpam-546	149	8	.	.	PUNCT
ejpam-546	150	1	j.	j.	PROPN
ejpam-546	150	2	pure	pure	PROPN
ejpam-546	150	3	appl	appl	PROPN
ejpam-546	150	4	.	.	PROPN
ejpam-546	150	5	math	math	PROPN
ejpam-546	150	6	,	,	PUNCT
ejpam-546	150	7	3	3	NUM
ejpam-546	150	8	(	(	PUNCT
ejpam-546	150	9	2010	2010	NUM
ejpam-546	150	10	)	)	PUNCT
ejpam-546	150	11	,	,	PUNCT
ejpam-546	150	12	118	118	NUM
ejpam-546	150	13	-	-	SYM
ejpam-546	150	14	127	127	NUM
ejpam-546	150	15	125	125	NUM
ejpam-546	150	16	=	=	SYM
ejpam-546	150	17	=	=	SYM
ejpam-546	150	18	=	=	SYM
ejpam-546	150	19	–	–	PUNCT
ejpam-546	150	20	(	(	PUNCT
ejpam-546	150	21	n+1	n+1	NOUN
ejpam-546	150	22	)	)	PUNCT
ejpam-546	150	23	figure	figure	NOUN
ejpam-546	150	24	11	11	NUM
ejpam-546	150	25	here	here	ADV
ejpam-546	150	26	we	we	PRON
ejpam-546	150	27	stack	stack	VERB
ejpam-546	150	28	layers	layer	NOUN
ejpam-546	150	29	of	of	ADP
ejpam-546	150	30	unit	unit	NOUN
ejpam-546	150	31	cubes	cube	NOUN
ejpam-546	150	32	to	to	PART
ejpam-546	150	33	represent	represent	VERB
ejpam-546	150	34	the	the	DET
ejpam-546	150	35	triangular	triangular	NOUN
ejpam-546	150	36	numbers	number	NOUN
ejpam-546	150	37	.	.	PUNCT
ejpam-546	151	1	the	the	DET
ejpam-546	151	2	sum	sum	NOUN
ejpam-546	151	3	of	of	ADP
ejpam-546	151	4	the	the	DET
ejpam-546	151	5	triangular	triangular	NOUN
ejpam-546	151	6	numbers	number	NOUN
ejpam-546	151	7	is	be	AUX
ejpam-546	151	8	total	total	ADJ
ejpam-546	151	9	number	number	NOUN
ejpam-546	151	10	of	of	ADP
ejpam-546	151	11	cubes	cube	NOUN
ejpam-546	151	12	,	,	PUNCT
ejpam-546	151	13	which	which	PRON
ejpam-546	151	14	is	be	AUX
ejpam-546	151	15	the	the	DET
ejpam-546	151	16	same	same	ADJ
ejpam-546	151	17	as	as	ADP
ejpam-546	151	18	the	the	DET
ejpam-546	151	19	total	total	ADJ
ejpam-546	151	20	volume	volume	NOUN
ejpam-546	151	21	of	of	ADP
ejpam-546	151	22	the	the	DET
ejpam-546	151	23	cubes	cube	NOUN
ejpam-546	151	24	.	.	PUNCT
ejpam-546	152	1	to	to	PART
ejpam-546	152	2	compute	compute	VERB
ejpam-546	152	3	the	the	DET
ejpam-546	152	4	volume	volume	NOUN
ejpam-546	152	5	,	,	PUNCT
ejpam-546	152	6	we	we	PRON
ejpam-546	152	7	“	"	PUNCT
ejpam-546	152	8	slice	slice	PROPN
ejpam-546	152	9	”	"	PUNCT
ejpam-546	152	10	off	off	ADP
ejpam-546	152	11	small	small	ADJ
ejpam-546	152	12	pyramids	pyramid	NOUN
ejpam-546	152	13	(	(	PUNCT
ejpam-546	152	14	shaded	shade	VERB
ejpam-546	152	15	gray	gray	ADJ
ejpam-546	152	16	)	)	PUNCT
ejpam-546	152	17	and	and	CCONJ
ejpam-546	152	18	place	place	VERB
ejpam-546	152	19	each	each	DET
ejpam-546	152	20	small	small	ADJ
ejpam-546	152	21	pyramid	pyramid	NOUN
ejpam-546	152	22	on	on	ADP
ejpam-546	152	23	the	the	DET
ejpam-546	152	24	top	top	NOUN
ejpam-546	152	25	of	of	ADP
ejpam-546	152	26	the	the	DET
ejpam-546	152	27	cube	cube	NOUN
ejpam-546	152	28	from	from	ADP
ejpam-546	152	29	which	which	PRON
ejpam-546	152	30	it	it	PRON
ejpam-546	152	31	came	come	VERB
ejpam-546	152	32	.	.	PUNCT
ejpam-546	153	1	the	the	DET
ejpam-546	153	2	result	result	NOUN
ejpam-546	153	3	is	be	AUX
ejpam-546	153	4	a	a	DET
ejpam-546	153	5	large	large	ADJ
ejpam-546	153	6	right	right	ADJ
ejpam-546	153	7	triangular	triangular	NOUN
ejpam-546	153	8	pyramid	pyramid	NOUN
ejpam-546	153	9	minus	minus	ADP
ejpam-546	153	10	some	some	DET
ejpam-546	153	11	smaller	small	ADJ
ejpam-546	153	12	right	right	ADJ
ejpam-546	153	13	triangular	triangular	NOUN
ejpam-546	153	14	pyramids	pyramid	NOUN
ejpam-546	153	15	along	along	ADP
ejpam-546	153	16	one	one	NUM
ejpam-546	153	17	edge	edge	NOUN
ejpam-546	153	18	of	of	ADP
ejpam-546	153	19	the	the	DET
ejpam-546	153	20	base	base	NOUN
ejpam-546	153	21	.	.	PUNCT
ejpam-546	154	1	thus	thus	ADV
ejpam-546	154	2	t1	t1	VERB
ejpam-546	154	3	+	+	CCONJ
ejpam-546	154	4	t2	t2	NOUN
ejpam-546	154	5	+	+	CCONJ
ejpam-546	154	6	t3	t3	PROPN
ejpam-546	154	7	+	+	X
ejpam-546	154	8	.	.	PUNCT
ejpam-546	154	9	.	.	PUNCT
ejpam-546	155	1	.+	.+	NOUN
ejpam-546	155	2	tn	tn	NOUN
ejpam-546	156	1	=	=	SYM
ejpam-546	156	2	1	1	NUM
ejpam-546	156	3	6	6	NUM
ejpam-546	156	4	(	(	PUNCT
ejpam-546	156	5	n+	n+	NUM
ejpam-546	156	6	1)3−	1)3−	NUM
ejpam-546	156	7	(	(	PUNCT
ejpam-546	156	8	n+	n+	NOUN
ejpam-546	156	9	1	1	NUM
ejpam-546	156	10	)	)	PUNCT
ejpam-546	156	11	1	1	NUM
ejpam-546	156	12	6	6	NUM
ejpam-546	156	13	=	=	SYM
ejpam-546	156	14	n(n+	n(n+	NOUN
ejpam-546	156	15	1)(n+	1)(n+	NUM
ejpam-546	156	16	2	2	NUM
ejpam-546	156	17	)	)	PUNCT
ejpam-546	156	18	6	6	NUM
ejpam-546	156	19	.	.	PUNCT
ejpam-546	157	1	in	in	ADP
ejpam-546	157	2	the	the	DET
ejpam-546	157	3	above	above	ADJ
ejpam-546	157	4	proof	proof	NOUN
ejpam-546	157	5	we	we	PRON
ejpam-546	157	6	evaluated	evaluate	VERB
ejpam-546	157	7	the	the	DET
ejpam-546	157	8	sum	sum	NOUN
ejpam-546	157	9	of	of	ADP
ejpam-546	157	10	the	the	DET
ejpam-546	157	11	first	first	ADJ
ejpam-546	157	12	n	n	PRON
ejpam-546	157	13	triangular	triangular	NOUN
ejpam-546	157	14	numbers	number	NOUN
ejpam-546	157	15	by	by	ADP
ejpam-546	157	16	computing	compute	VERB
ejpam-546	157	17	volumes	volume	NOUN
ejpam-546	157	18	of	of	ADP
ejpam-546	157	19	pyramids	pyramid	NOUN
ejpam-546	157	20	.	.	PUNCT
ejpam-546	158	1	this	this	PRON
ejpam-546	158	2	is	be	AUX
ejpam-546	158	3	actually	actually	ADV
ejpam-546	158	4	an	an	DET
ejpam-546	158	5	extension	extension	NOUN
ejpam-546	158	6	of	of	ADP
ejpam-546	158	7	the	the	DET
ejpam-546	158	8	fubini	fubini	ADJ
ejpam-546	158	9	principle	principle	NOUN
ejpam-546	158	10	from	from	ADP
ejpam-546	158	11	simple	simple	ADJ
ejpam-546	158	12	enumeration	enumeration	NOUN
ejpam-546	158	13	of	of	ADP
ejpam-546	158	14	objects	object	NOUN
ejpam-546	158	15	to	to	PART
ejpam-546	158	16	additive	additive	VERB
ejpam-546	158	17	measures	measure	NOUN
ejpam-546	158	18	such	such	ADJ
ejpam-546	158	19	as	as	ADP
ejpam-546	158	20	length	length	NOUN
ejpam-546	158	21	,	,	PUNCT
ejpam-546	158	22	area	area	NOUN
ejpam-546	158	23	and	and	CCONJ
ejpam-546	158	24	volume	volume	NOUN
ejpam-546	158	25	.	.	PUNCT
ejpam-546	159	1	the	the	DET
ejpam-546	159	2	volume	volume	NOUN
ejpam-546	159	3	version	version	NOUN
ejpam-546	159	4	of	of	ADP
ejpam-546	159	5	the	the	DET
ejpam-546	159	6	fubini	fubini	ADJ
ejpam-546	159	7	principle	principle	NOUN
ejpam-546	159	8	is	be	AUX
ejpam-546	159	9	:	:	PUNCT
ejpam-546	159	10	computing	compute	VERB
ejpam-546	159	11	the	the	DET
ejpam-546	159	12	volume	volume	NOUN
ejpam-546	159	13	of	of	ADP
ejpam-546	159	14	an	an	DET
ejpam-546	159	15	object	object	NOUN
ejpam-546	159	16	in	in	ADP
ejpam-546	159	17	two	two	NUM
ejpam-546	159	18	different	different	ADJ
ejpam-546	159	19	ways	way	NOUN
ejpam-546	159	20	yields	yield	VERB
ejpam-546	159	21	the	the	DET
ejpam-546	159	22	same	same	ADJ
ejpam-546	159	23	number	number	NOUN
ejpam-546	159	24	;	;	PUNCT
ejpam-546	159	25	and	and	CCONJ
ejpam-546	159	26	similarly	similarly	ADV
ejpam-546	159	27	for	for	ADP
ejpam-546	159	28	length	length	NOUN
ejpam-546	159	29	and	and	CCONJ
ejpam-546	159	30	area	area	NOUN
ejpam-546	159	31	.	.	PUNCT
ejpam-546	160	1	we	we	PRON
ejpam-546	160	2	can	can	AUX
ejpam-546	160	3	not	not	PART
ejpam-546	160	4	,	,	PUNCT
ejpam-546	160	5	however	however	ADV
ejpam-546	160	6	,	,	PUNCT
ejpam-546	160	7	extend	extend	VERB
ejpam-546	160	8	the	the	DET
ejpam-546	160	9	cantor	cantor	NOUN
ejpam-546	160	10	principle	principle	NOUN
ejpam-546	160	11	to	to	AUX
ejpam-546	160	12	additive	additive	VERB
ejpam-546	160	13	measures	measure	NOUN
ejpam-546	160	14	for	for	ADP
ejpam-546	160	15	example	example	NOUN
ejpam-546	160	16	,	,	PUNCT
ejpam-546	160	17	one	one	PRON
ejpam-546	160	18	can	can	AUX
ejpam-546	160	19	construct	construct	VERB
ejpam-546	160	20	a	a	DET
ejpam-546	160	21	one	one	NUM
ejpam-546	160	22	-	-	PUNCT
ejpam-546	160	23	to	to	ADP
ejpam-546	160	24	-	-	PUNCT
ejpam-546	160	25	one	one	NUM
ejpam-546	160	26	correspondence	correspondence	NOUN
ejpam-546	160	27	between	between	ADP
ejpam-546	160	28	the	the	DET
ejpam-546	160	29	points	point	NOUN
ejpam-546	160	30	on	on	ADP
ejpam-546	160	31	two	two	NUM
ejpam-546	160	32	line	line	NOUN
ejpam-546	160	33	segments	segment	NOUN
ejpam-546	160	34	with	with	ADP
ejpam-546	160	35	different	different	ADJ
ejpam-546	160	36	lengths	length	NOUN
ejpam-546	160	37	.	.	PUNCT
ejpam-546	161	1	theorem	theorem	VERB
ejpam-546	161	2	9	9	NUM
ejpam-546	161	3	.	.	PUNCT
ejpam-546	161	4	for	for	ADP
ejpam-546	161	5	all	all	DET
ejpam-546	161	6	n≥	n≥	ADJ
ejpam-546	161	7	1	1	NUM
ejpam-546	161	8	,	,	PUNCT
ejpam-546	161	9	13	13	NUM
ejpam-546	162	1	+	+	SYM
ejpam-546	162	2	23	23	NUM
ejpam-546	162	3	+	+	NUM
ejpam-546	162	4	33	33	NUM
ejpam-546	163	1	+	+	NUM
ejpam-546	163	2	.	.	PUNCT
ejpam-546	164	1	.	.	PUNCT
ejpam-546	165	1	.+	.+	NOUN
ejpam-546	165	2	n3	n3	NOUN
ejpam-546	165	3	=	=	SYM
ejpam-546	165	4	(	(	PUNCT
ejpam-546	165	5	1	1	NUM
ejpam-546	165	6	+	+	NUM
ejpam-546	165	7	2	2	NUM
ejpam-546	165	8	+	+	NUM
ejpam-546	165	9	3	3	NUM
ejpam-546	165	10	+	+	NUM
ejpam-546	165	11	.	.	PUNCT
ejpam-546	165	12	.	.	PUNCT
ejpam-546	166	1	.+	.+	NOUN
ejpam-546	166	2	n)2	n)2	NOUN
ejpam-546	166	3	=	=	SYM
ejpam-546	166	4	t2	t2	PROPN
ejpam-546	166	5	n.	n.	NOUN
ejpam-546	166	6	proof	proof	NOUN
ejpam-546	166	7	.	.	PUNCT
ejpam-546	167	1	again	again	ADV
ejpam-546	167	2	,	,	PUNCT
ejpam-546	167	3	we	we	PRON
ejpam-546	167	4	give	give	VERB
ejpam-546	167	5	two	two	NUM
ejpam-546	167	6	proofs	proof	NOUN
ejpam-546	167	7	.	.	PUNCT
ejpam-546	168	1	in	in	ADP
ejpam-546	168	2	the	the	DET
ejpam-546	168	3	first	first	ADJ
ejpam-546	168	4	,	,	PUNCT
ejpam-546	168	5	we	we	PRON
ejpam-546	168	6	represent	represent	VERB
ejpam-546	168	7	k3	k3	VERB
ejpam-546	168	8	as	as	SCONJ
ejpam-546	168	9	k	k	PROPN
ejpam-546	168	10	copies	copy	NOUN
ejpam-546	168	11	of	of	ADP
ejpam-546	168	12	a	a	DET
ejpam-546	168	13	square	square	NOUN
ejpam-546	168	14	with	with	ADP
ejpam-546	168	15	area	area	NOUN
ejpam-546	168	16	k2	k2	PROPN
ejpam-546	168	17	to	to	PART
ejpam-546	168	18	establish	establish	VERB
ejpam-546	168	19	the	the	DET
ejpam-546	168	20	identity	identity	NOUN
ejpam-546	168	21	[	[	X
ejpam-546	168	22	3	3	NUM
ejpam-546	168	23	,	,	PUNCT
ejpam-546	168	24	7	7	NUM
ejpam-546	168	25	]	]	PUNCT
ejpam-546	168	26	.	.	PUNCT
ejpam-546	169	1	figure	figure	NOUN
ejpam-546	169	2	12	12	NUM
ejpam-546	169	3	in	in	ADP
ejpam-546	169	4	figure	figure	NOUN
ejpam-546	169	5	12	12	NUM
ejpam-546	169	6	,	,	PUNCT
ejpam-546	169	7	we	we	PRON
ejpam-546	169	8	have	have	VERB
ejpam-546	169	9	4(13	4(13	NUM
ejpam-546	169	10	+	+	NOUN
ejpam-546	169	11	23	23	NUM
ejpam-546	169	12	+	+	NUM
ejpam-546	169	13	33	33	NUM
ejpam-546	169	14	+	+	NUM
ejpam-546	169	15	.	.	PUNCT
ejpam-546	169	16	.	.	PUNCT
ejpam-546	170	1	.+	.+	NOUN
ejpam-546	170	2	n3	n3	NOUN
ejpam-546	170	3	)	)	PUNCT
ejpam-546	170	4	=	=	PUNCT
ejpam-546	171	1	[	[	X
ejpam-546	171	2	n(n+	n(n+	NUM
ejpam-546	171	3	1)]2	1)]2	NOUN
ejpam-546	171	4	(	(	PUNCT
ejpam-546	171	5	for	for	ADP
ejpam-546	171	6	n=	n=	ADJ
ejpam-546	171	7	4	4	NUM
ejpam-546	171	8	)	)	PUNCT
ejpam-546	171	9	.	.	PUNCT
ejpam-546	172	1	for	for	ADP
ejpam-546	172	2	the	the	DET
ejpam-546	172	3	second	second	ADJ
ejpam-546	172	4	proof	proof	NOUN
ejpam-546	172	5	,	,	PUNCT
ejpam-546	172	6	we	we	PRON
ejpam-546	172	7	use	use	VERB
ejpam-546	172	8	the	the	DET
ejpam-546	172	9	fact	fact	NOUN
ejpam-546	172	10	that	that	SCONJ
ejpam-546	172	11	1	1	NUM
ejpam-546	172	12	+	+	SYM
ejpam-546	172	13	2	2	NUM
ejpam-546	172	14	+	+	NOUN
ejpam-546	172	15	3	3	NUM
ejpam-546	172	16	+	+	NOUN
ejpam-546	172	17	.	.	PUNCT
ejpam-546	172	18	.	.	PUNCT
ejpam-546	173	1	.+(n−1)+n+(n−1)+	.+(n−1)+n+(n−1)+	PROPN
ejpam-546	173	2	.	.	PUNCT
ejpam-546	173	3	.	.	PUNCT
ejpam-546	174	1	.+2	.+2	PUNCT
ejpam-546	175	1	+	+	SYM
ejpam-546	175	2	1=	1=	X
ejpam-546	175	3	n2	n2	NOUN
ejpam-546	175	4	(	(	PUNCT
ejpam-546	175	5	we	we	PRON
ejpam-546	175	6	leave	leave	VERB
ejpam-546	175	7	it	it	PRON
ejpam-546	175	8	as	as	ADP
ejpam-546	175	9	an	an	DET
ejpam-546	175	10	exercise	exercise	NOUN
ejpam-546	175	11	for	for	SCONJ
ejpam-546	175	12	the	the	DET
ejpam-546	175	13	reader	reader	NOUN
ejpam-546	175	14	to	to	PART
ejpam-546	175	15	draw	draw	VERB
ejpam-546	175	16	a	a	DET
ejpam-546	175	17	picture	picture	NOUN
ejpam-546	175	18	with	with	ADP
ejpam-546	175	19	balls	ball	NOUN
ejpam-546	175	20	in	in	ADP
ejpam-546	175	21	a	a	DET
ejpam-546	175	22	square	square	ADJ
ejpam-546	175	23	array	array	NOUN
ejpam-546	175	24	and	and	CCONJ
ejpam-546	175	25	c.	c.	PROPN
ejpam-546	175	26	alsina	alsina	PROPN
ejpam-546	175	27	and	and	CCONJ
ejpam-546	175	28	r.	r.	PROPN
ejpam-546	175	29	nelsen	nelsen	PROPN
ejpam-546	175	30	/	/	SYM
ejpam-546	175	31	eur	eur	PROPN
ejpam-546	175	32	.	.	PUNCT
ejpam-546	176	1	j.	j.	PROPN
ejpam-546	176	2	pure	pure	PROPN
ejpam-546	176	3	appl	appl	PROPN
ejpam-546	176	4	.	.	PROPN
ejpam-546	176	5	math	math	PROPN
ejpam-546	176	6	,	,	PUNCT
ejpam-546	176	7	3	3	NUM
ejpam-546	176	8	(	(	PUNCT
ejpam-546	176	9	2010	2010	NUM
ejpam-546	176	10	)	)	PUNCT
ejpam-546	176	11	,	,	PUNCT
ejpam-546	176	12	118	118	NUM
ejpam-546	176	13	-	-	SYM
ejpam-546	176	14	127	127	NUM
ejpam-546	176	15	126	126	NUM
ejpam-546	176	16	count	count	NOUN
ejpam-546	176	17	the	the	DET
ejpam-546	176	18	balls	ball	NOUN
ejpam-546	176	19	by	by	ADP
ejpam-546	176	20	diagonals	diagonal	NOUN
ejpam-546	176	21	in	in	ADP
ejpam-546	176	22	the	the	DET
ejpam-546	176	23	square	square	ADJ
ejpam-546	176	24	array	array	NOUN
ejpam-546	176	25	)	)	PUNCT
ejpam-546	176	26	and	and	CCONJ
ejpam-546	176	27	consider	consider	VERB
ejpam-546	176	28	a	a	DET
ejpam-546	176	29	square	square	ADJ
ejpam-546	176	30	array	array	NOUN
ejpam-546	176	31	of	of	ADP
ejpam-546	176	32	numbers	number	NOUN
ejpam-546	176	33	(	(	PUNCT
ejpam-546	176	34	rather	rather	ADV
ejpam-546	176	35	than	than	ADP
ejpam-546	176	36	balls	ball	NOUN
ejpam-546	176	37	)	)	PUNCT
ejpam-546	176	38	in	in	ADP
ejpam-546	176	39	which	which	PRON
ejpam-546	176	40	the	the	DET
ejpam-546	176	41	element	element	NOUN
ejpam-546	176	42	in	in	ADP
ejpam-546	176	43	row	row	NOUN
ejpam-546	176	44	i	i	PRON
ejpam-546	176	45	and	and	CCONJ
ejpam-546	176	46	column	column	PROPN
ejpam-546	176	47	j	j	PROPN
ejpam-546	176	48	is	be	AUX
ejpam-546	176	49	i	i	PROPN
ejpam-546	176	50	j	j	PROPN
ejpam-546	176	51	,	,	PUNCT
ejpam-546	176	52	and	and	CCONJ
ejpam-546	176	53	sum	sum	VERB
ejpam-546	176	54	the	the	DET
ejpam-546	176	55	numbers	number	NOUN
ejpam-546	176	56	in	in	ADP
ejpam-546	176	57	two	two	NUM
ejpam-546	176	58	different	different	ADJ
ejpam-546	176	59	ways	way	NOUN
ejpam-546	176	60	.	.	PUNCT
ejpam-546	177	1	see	see	VERB
ejpam-546	177	2	figure	figure	NOUN
ejpam-546	177	3	13	13	NUM
ejpam-546	178	1	[	[	X
ejpam-546	178	2	10	10	NUM
ejpam-546	178	3	]	]	PUNCT
ejpam-546	178	4	.	.	PUNCT
ejpam-546	178	5	1	1	NUM
ejpam-546	178	6	2	2	NUM
ejpam-546	178	7	3	3	NUM
ejpam-546	178	8	n	n	NUM
ejpam-546	178	9	2	2	NUM
ejpam-546	178	10	4	4	NUM
ejpam-546	178	11	6	6	NUM
ejpam-546	178	12	2n	2n	NUM
ejpam-546	178	13	3	3	NUM
ejpam-546	178	14	6	6	NUM
ejpam-546	178	15	9	9	NUM
ejpam-546	178	16	3n	3n	NUM
ejpam-546	178	17	n	n	CCONJ
ejpam-546	178	18	2n	2n	NUM
ejpam-546	178	19	3n	3n	NUM
ejpam-546	178	20	n2	n2	NOUN
ejpam-546	178	21	...	...	PUNCT
ejpam-546	178	22	...	...	PUNCT
ejpam-546	178	23	...	...	PUNCT
ejpam-546	178	24	...	...	PUNCT
ejpam-546	178	25	..	..	PUNCT
ejpam-546	178	26	.	.	PUNCT
ejpam-546	178	27	...	...	PUNCT
ejpam-546	179	1	...	...	PUNCT
ejpam-546	179	2	...	...	PUNCT
ejpam-546	180	1	1	1	NUM
ejpam-546	180	2	2	2	NUM
ejpam-546	180	3	3	3	NUM
ejpam-546	180	4	n	n	NUM
ejpam-546	180	5	2	2	NUM
ejpam-546	180	6	4	4	NUM
ejpam-546	180	7	6	6	NUM
ejpam-546	180	8	2n	2n	NUM
ejpam-546	180	9	3	3	NUM
ejpam-546	180	10	6	6	NUM
ejpam-546	180	11	9	9	NUM
ejpam-546	180	12	3n	3n	NUM
ejpam-546	180	13	n	n	CCONJ
ejpam-546	180	14	2n	2n	NUM
ejpam-546	180	15	3n	3n	NUM
ejpam-546	180	16	n2	n2	NOUN
ejpam-546	180	17	...	...	PUNCT
ejpam-546	180	18	...	...	PUNCT
ejpam-546	180	19	...	...	PUNCT
ejpam-546	180	20	...	...	PUNCT
ejpam-546	180	21	..	..	PUNCT
ejpam-546	180	22	.	.	PUNCT
ejpam-546	180	23	...	...	PUNCT
ejpam-546	180	24	...	...	PUNCT
ejpam-546	180	25	...	...	PUNCT
ejpam-546	181	1	figure	figure	NOUN
ejpam-546	181	2	13	13	NUM
ejpam-546	181	3	summing	sum	VERB
ejpam-546	181	4	by	by	ADP
ejpam-546	181	5	columns	column	NOUN
ejpam-546	181	6	yields	yield	NOUN
ejpam-546	181	7	n	n	INTJ
ejpam-546	181	8	∑	∑	PROPN
ejpam-546	181	9	i=1	i=1	PROPN
ejpam-546	182	1	i	i	PRON
ejpam-546	182	2	+	+	CCONJ
ejpam-546	182	3	2	2	NUM
ejpam-546	182	4	n	n	NOUN
ejpam-546	182	5	∑	∑	NOUN
ejpam-546	182	6	i=1	i=1	PROPN
ejpam-546	182	7	i	i	PRON
ejpam-546	182	8	!	!	PUNCT
ejpam-546	183	1	+	+	CCONJ
ejpam-546	183	2	.	.	PUNCT
ejpam-546	183	3	.	.	PUNCT
ejpam-546	184	1	.+	.+	NOUN
ejpam-546	184	2	n	n	CCONJ
ejpam-546	184	3	n	n	NOUN
ejpam-546	184	4	∑	∑	PUNCT
ejpam-546	184	5	i=1	i=1	PROPN
ejpam-546	184	6	i	i	PRON
ejpam-546	184	7	!	!	PUNCT
ejpam-546	185	1	=	=	PUNCT
ejpam-546	186	1	n	n	CCONJ
ejpam-546	186	2	∑	∑	NOUN
ejpam-546	186	3	i=1	i=1	PROPN
ejpam-546	187	1	i	i	PRON
ejpam-546	187	2	!	!	PUNCT
ejpam-546	187	3	2	2	NUM
ejpam-546	187	4	,	,	PUNCT
ejpam-546	187	5	while	while	SCONJ
ejpam-546	187	6	summing	sum	VERB
ejpam-546	187	7	by	by	ADP
ejpam-546	187	8	the	the	DET
ejpam-546	187	9	l	l	NOUN
ejpam-546	187	10	-	-	ADJ
ejpam-546	187	11	shaped	shape	VERB
ejpam-546	187	12	shaded	shade	VERB
ejpam-546	187	13	regions	region	NOUN
ejpam-546	187	14	yields	yield	VERB
ejpam-546	187	15	1	1	NUM
ejpam-546	187	16	·	·	SYM
ejpam-546	187	17	12	12	NUM
ejpam-546	187	18	+	+	SYM
ejpam-546	187	19	2	2	NUM
ejpam-546	187	20	·	·	SYM
ejpam-546	187	21	22	22	NUM
ejpam-546	188	1	+	+	CCONJ
ejpam-546	188	2	.	.	PUNCT
ejpam-546	188	3	.	.	PUNCT
ejpam-546	189	1	.+	.+	NOUN
ejpam-546	189	2	n	n	CCONJ
ejpam-546	189	3	·	·	PUNCT
ejpam-546	189	4	n2	n2	ADJ
ejpam-546	189	5	+	+	X
ejpam-546	189	6	=	=	SYM
ejpam-546	189	7	∑n	∑n	PROPN
ejpam-546	189	8	i=1	i=1	PROPN
ejpam-546	189	9	i3	i3	NOUN
ejpam-546	189	10	.	.	PUNCT
ejpam-546	190	1	we	we	PRON
ejpam-546	190	2	conclude	conclude	VERB
ejpam-546	190	3	this	this	DET
ejpam-546	190	4	section	section	NOUN
ejpam-546	190	5	with	with	ADP
ejpam-546	190	6	a	a	DET
ejpam-546	190	7	theorem	theorem	NOUN
ejpam-546	190	8	representing	represent	VERB
ejpam-546	190	9	a	a	DET
ejpam-546	190	10	cube	cube	NOUN
ejpam-546	190	11	as	as	ADP
ejpam-546	190	12	a	a	DET
ejpam-546	190	13	double	double	ADJ
ejpam-546	190	14	sum	sum	NOUN
ejpam-546	190	15	of	of	ADP
ejpam-546	190	16	integers	integer	NOUN
ejpam-546	190	17	.	.	PUNCT
ejpam-546	191	1	theorem	theorem	ADJ
ejpam-546	191	2	10	10	NUM
ejpam-546	191	3	.	.	PUNCT
ejpam-546	192	1	for	for	ADP
ejpam-546	192	2	all	all	DET
ejpam-546	192	3	n≥	n≥	ADJ
ejpam-546	192	4	1	1	NUM
ejpam-546	192	5	,	,	PUNCT
ejpam-546	192	6	∑n	∑n	PROPN
ejpam-546	192	7	i=1	i=1	PROPN
ejpam-546	192	8	∑n	∑n	PROPN
ejpam-546	192	9	j=1(i+	j=1(i+	NOUN
ejpam-546	192	10	j+	j+	NUM
ejpam-546	192	11	1	1	NUM
ejpam-546	192	12	)	)	PUNCT
ejpam-546	192	13	=	=	SYM
ejpam-546	192	14	n3	n3	VERB
ejpam-546	192	15	.	.	PROPN
ejpam-546	192	16	proof	proof	NOUN
ejpam-546	192	17	.	.	PUNCT
ejpam-546	193	1	we	we	PRON
ejpam-546	193	2	represent	represent	VERB
ejpam-546	193	3	the	the	DET
ejpam-546	193	4	double	double	ADJ
ejpam-546	193	5	sum	sum	NOUN
ejpam-546	193	6	as	as	ADP
ejpam-546	193	7	a	a	DET
ejpam-546	193	8	collection	collection	NOUN
ejpam-546	193	9	of	of	ADP
ejpam-546	193	10	unit	unit	NOUN
ejpam-546	193	11	cubes	cube	NOUN
ejpam-546	193	12	and	and	CCONJ
ejpam-546	193	13	compute	compute	VERB
ejpam-546	193	14	the	the	DET
ejpam-546	193	15	volume	volume	NOUN
ejpam-546	193	16	of	of	ADP
ejpam-546	193	17	a	a	DET
ejpam-546	193	18	rectangular	rectangular	ADJ
ejpam-546	193	19	box	box	NOUN
ejpam-546	193	20	composed	compose	VERB
ejpam-546	193	21	of	of	ADP
ejpam-546	193	22	two	two	NUM
ejpam-546	193	23	copies	copy	NOUN
ejpam-546	193	24	of	of	ADP
ejpam-546	193	25	the	the	DET
ejpam-546	193	26	collection	collection	NOUN
ejpam-546	193	27	.	.	PUNCT
ejpam-546	194	1	see	see	VERB
ejpam-546	194	2	figure	figure	NOUN
ejpam-546	194	3	14	14	NUM
ejpam-546	194	4	.	.	PUNCT
ejpam-546	195	1	figure	figure	NOUN
ejpam-546	195	2	14	14	NUM
ejpam-546	195	3	observe	observe	VERB
ejpam-546	195	4	that	that	SCONJ
ejpam-546	195	5	two	two	NUM
ejpam-546	195	6	copies	copy	NOUN
ejpam-546	195	7	of	of	ADP
ejpam-546	195	8	the	the	DET
ejpam-546	195	9	sum	sum	NOUN
ejpam-546	195	10	s	s	PART
ejpam-546	195	11	=	=	PUNCT
ejpam-546	195	12	∑n	∑n	PROPN
ejpam-546	195	13	i=1	i=1	PROPN
ejpam-546	195	14	∑n	∑n	PROPN
ejpam-546	195	15	j=1(i+	j=1(i+	PROPN
ejpam-546	195	16	j+1	j+1	NUM
ejpam-546	195	17	)	)	PUNCT
ejpam-546	195	18	fit	fit	ADJ
ejpam-546	195	19	into	into	ADP
ejpam-546	195	20	a	a	DET
ejpam-546	195	21	rectangular	rectangular	ADJ
ejpam-546	195	22	box	box	NOUN
ejpam-546	195	23	with	with	ADP
ejpam-546	195	24	base	base	NOUN
ejpam-546	195	25	n2	n2	NOUN
ejpam-546	195	26	and	and	CCONJ
ejpam-546	195	27	height	height	NOUN
ejpam-546	195	28	2n	2n	NUM
ejpam-546	195	29	,	,	PUNCT
ejpam-546	195	30	hence	hence	ADV
ejpam-546	195	31	computing	compute	VERB
ejpam-546	195	32	the	the	DET
ejpam-546	195	33	volume	volume	NOUN
ejpam-546	195	34	of	of	ADP
ejpam-546	195	35	the	the	DET
ejpam-546	195	36	box	box	NOUN
ejpam-546	195	37	in	in	ADP
ejpam-546	195	38	two	two	NUM
ejpam-546	195	39	ways	way	NOUN
ejpam-546	195	40	yields	yield	VERB
ejpam-546	195	41	2s	2s	NUM
ejpam-546	195	42	=	=	SYM
ejpam-546	195	43	2n3	2n3	NUM
ejpam-546	195	44	,	,	PUNCT
ejpam-546	195	45	or	or	CCONJ
ejpam-546	195	46	s	s	NOUN
ejpam-546	195	47	=	=	SYM
ejpam-546	195	48	n3	n3	NOUN
ejpam-546	195	49	.	.	PROPN
ejpam-546	195	50	5	5	NUM
ejpam-546	195	51	.	.	X
ejpam-546	195	52	conclusion	conclusion	NOUN
ejpam-546	195	53	in	in	ADP
ejpam-546	195	54	this	this	DET
ejpam-546	195	55	short	short	ADJ
ejpam-546	195	56	survey	survey	NOUN
ejpam-546	195	57	we	we	PRON
ejpam-546	195	58	have	have	AUX
ejpam-546	195	59	looked	look	VERB
ejpam-546	195	60	at	at	ADP
ejpam-546	195	61	visual	visual	ADJ
ejpam-546	195	62	proofs	proof	NOUN
ejpam-546	195	63	using	use	VERB
ejpam-546	195	64	representations	representation	NOUN
ejpam-546	195	65	of	of	ADP
ejpam-546	195	66	numbers	number	NOUN
ejpam-546	195	67	by	by	ADP
ejpam-546	195	68	sets	set	NOUN
ejpam-546	195	69	of	of	ADP
ejpam-546	195	70	objects	object	NOUN
ejpam-546	195	71	.	.	PUNCT
ejpam-546	196	1	of	of	ADP
ejpam-546	196	2	course	course	NOUN
ejpam-546	196	3	,	,	PUNCT
ejpam-546	196	4	there	there	PRON
ejpam-546	196	5	are	be	VERB
ejpam-546	196	6	many	many	ADJ
ejpam-546	196	7	other	other	ADJ
ejpam-546	196	8	ways	way	NOUN
ejpam-546	196	9	to	to	PART
ejpam-546	196	10	represent	represent	VERB
ejpam-546	196	11	numbers	number	NOUN
ejpam-546	196	12	(	(	PUNCT
ejpam-546	196	13	not	not	PART
ejpam-546	196	14	necessarily	necessarily	ADV
ejpam-546	196	15	references	reference	VERB
ejpam-546	196	16	127	127	NUM
ejpam-546	196	17	natural	natural	ADJ
ejpam-546	196	18	numbers	number	NOUN
ejpam-546	196	19	)	)	PUNCT
ejpam-546	196	20	,	,	PUNCT
ejpam-546	196	21	including	include	VERB
ejpam-546	196	22	representing	represent	VERB
ejpam-546	196	23	them	they	PRON
ejpam-546	196	24	as	as	ADP
ejpam-546	196	25	lengths	length	NOUN
ejpam-546	196	26	of	of	ADP
ejpam-546	196	27	segments	segment	NOUN
ejpam-546	196	28	,	,	PUNCT
ejpam-546	196	29	areas	area	NOUN
ejpam-546	196	30	of	of	ADP
ejpam-546	196	31	plane	plane	NOUN
ejpam-546	196	32	figures	figure	NOUN
ejpam-546	196	33	,	,	PUNCT
ejpam-546	196	34	volumes	volume	NOUN
ejpam-546	196	35	of	of	ADP
ejpam-546	196	36	objects	object	NOUN
ejpam-546	196	37	.	.	PUNCT
ejpam-546	197	1	with	with	ADP
ejpam-546	197	2	such	such	ADJ
ejpam-546	197	3	representations	representation	NOUN
ejpam-546	197	4	,	,	PUNCT
ejpam-546	197	5	other	other	ADJ
ejpam-546	197	6	techniques	technique	NOUN
ejpam-546	197	7	can	can	AUX
ejpam-546	197	8	be	be	AUX
ejpam-546	197	9	employed	employ	VERB
ejpam-546	197	10	in	in	ADP
ejpam-546	197	11	the	the	DET
ejpam-546	197	12	proofs	proof	NOUN
ejpam-546	197	13	,	,	PUNCT
ejpam-546	197	14	such	such	ADJ
ejpam-546	197	15	as	as	ADP
ejpam-546	197	16	rotations	rotation	NOUN
ejpam-546	197	17	,	,	PUNCT
ejpam-546	197	18	translations	translation	NOUN
ejpam-546	197	19	,	,	PUNCT
ejpam-546	197	20	reflections	reflection	NOUN
ejpam-546	197	21	,	,	PUNCT
ejpam-546	197	22	and	and	CCONJ
ejpam-546	197	23	other	other	ADJ
ejpam-546	197	24	transformations	transformation	NOUN
ejpam-546	197	25	that	that	PRON
ejpam-546	197	26	preserve	preserve	VERB
ejpam-546	197	27	length	length	NOUN
ejpam-546	197	28	,	,	PUNCT
ejpam-546	197	29	area	area	NOUN
ejpam-546	197	30	,	,	PUNCT
ejpam-546	197	31	or	or	CCONJ
ejpam-546	197	32	volume	volume	NOUN
ejpam-546	197	33	.	.	PUNCT
ejpam-546	198	1	for	for	ADP
ejpam-546	198	2	a	a	DET
ejpam-546	198	3	comprehensive	comprehensive	ADJ
ejpam-546	198	4	survey	survey	NOUN
ejpam-546	198	5	of	of	ADP
ejpam-546	198	6	these	these	DET
ejpam-546	198	7	techniques	technique	NOUN
ejpam-546	198	8	(	(	PUNCT
ejpam-546	198	9	and	and	CCONJ
ejpam-546	198	10	many	many	ADJ
ejpam-546	198	11	others	other	NOUN
ejpam-546	198	12	)	)	PUNCT
ejpam-546	198	13	,	,	PUNCT
ejpam-546	198	14	see	see	VERB
ejpam-546	198	15	the	the	DET
ejpam-546	198	16	books	book	NOUN
ejpam-546	198	17	math	math	NOUN
ejpam-546	198	18	made	make	VERB
ejpam-546	198	19	visual	visual	ADJ
ejpam-546	198	20	:	:	PUNCT
ejpam-546	198	21	creating	create	VERB
ejpam-546	198	22	images	image	NOUN
ejpam-546	198	23	for	for	ADP
ejpam-546	198	24	understanding	understand	VERB
ejpam-546	198	25	mathematics	mathematic	NOUN
ejpam-546	198	26	and	and	CCONJ
ejpam-546	198	27	when	when	SCONJ
ejpam-546	198	28	less	less	ADV
ejpam-546	198	29	is	be	AUX
ejpam-546	198	30	more	more	ADJ
ejpam-546	198	31	:	:	PUNCT
ejpam-546	198	32	visualizing	visualize	VERB
ejpam-546	198	33	basic	basic	ADJ
ejpam-546	198	34	inequalities	inequality	NOUN
ejpam-546	198	35	[	[	X
ejpam-546	198	36	1	1	NUM
ejpam-546	198	37	,	,	PUNCT
ejpam-546	198	38	2	2	NUM
ejpam-546	198	39	]	]	PUNCT
ejpam-546	198	40	.	.	PUNCT
ejpam-546	199	1	references	reference	NOUN
ejpam-546	199	2	[	[	X
ejpam-546	199	3	1	1	NUM
ejpam-546	199	4	]	]	X
ejpam-546	199	5	c	c	X
ejpam-546	199	6	alsina	alsina	NOUN
ejpam-546	199	7	and	and	CCONJ
ejpam-546	199	8	r	r	PROPN
ejpam-546	199	9	nelsen	nelsen	PROPN
ejpam-546	199	10	.	.	PUNCT
ejpam-546	200	1	math	math	PROPN
ejpam-546	200	2	made	make	VERB
ejpam-546	200	3	visual	visual	ADJ
ejpam-546	200	4	:	:	PUNCT
ejpam-546	200	5	creating	create	VERB
ejpam-546	200	6	images	image	NOUN
ejpam-546	200	7	for	for	ADP
ejpam-546	200	8	understanding	understand	VERB
ejpam-546	200	9	mathematics	mathematic	NOUN
ejpam-546	200	10	.	.	PUNCT
ejpam-546	201	1	mathematical	mathematical	ADJ
ejpam-546	201	2	association	association	PROPN
ejpam-546	201	3	of	of	ADP
ejpam-546	201	4	america	america	PROPN
ejpam-546	201	5	,	,	PUNCT
ejpam-546	201	6	washington	washington	PROPN
ejpam-546	201	7	,	,	PUNCT
ejpam-546	201	8	2006	2006	NUM
ejpam-546	201	9	.	.	PUNCT
ejpam-546	202	1	[	[	X
ejpam-546	202	2	2	2	NUM
ejpam-546	202	3	]	]	X
ejpam-546	202	4	c	c	X
ejpam-546	202	5	alsina	alsina	NOUN
ejpam-546	202	6	and	and	CCONJ
ejpam-546	202	7	r	r	PROPN
ejpam-546	202	8	nelsen	nelsen	PROPN
ejpam-546	202	9	.	.	PUNCT
ejpam-546	203	1	when	when	SCONJ
ejpam-546	203	2	less	less	ADJ
ejpam-546	203	3	is	be	AUX
ejpam-546	203	4	more	more	ADJ
ejpam-546	203	5	:	:	PUNCT
ejpam-546	203	6	visualizing	visualize	VERB
ejpam-546	203	7	basic	basic	ADJ
ejpam-546	203	8	inequalities	inequality	NOUN
ejpam-546	203	9	.	.	PUNCT
ejpam-546	204	1	mathematical	mathematical	PROPN
ejpam-546	204	2	association	association	PROPN
ejpam-546	204	3	of	of	ADP
ejpam-546	204	4	america	america	PROPN
ejpam-546	204	5	,	,	PUNCT
ejpam-546	204	6	washington	washington	PROPN
ejpam-546	204	7	,	,	PUNCT
ejpam-546	204	8	2009	2009	NUM
ejpam-546	204	9	.	.	PUNCT
ejpam-546	205	1	[	[	X
ejpam-546	205	2	3	3	X
ejpam-546	205	3	]	]	PUNCT
ejpam-546	205	4	a	a	DET
ejpam-546	205	5	cupillari	cupillari	NOUN
ejpam-546	205	6	.	.	PUNCT
ejpam-546	206	1	proof	proof	NOUN
ejpam-546	206	2	without	without	ADP
ejpam-546	206	3	words	word	NOUN
ejpam-546	206	4	.	.	PUNCT
ejpam-546	207	1	mathematics	mathematic	NOUN
ejpam-546	207	2	magazine	magazine	PROPN
ejpam-546	207	3	,	,	PUNCT
ejpam-546	207	4	(	(	PUNCT
ejpam-546	207	5	62):259	62):259	NOUN
ejpam-546	207	6	,	,	PUNCT
ejpam-546	207	7	1989	1989	NUM
ejpam-546	207	8	.	.	PUNCT
ejpam-546	208	1	[	[	X
ejpam-546	208	2	4	4	NUM
ejpam-546	208	3	]	]	X
ejpam-546	208	4	m	m	PROPN
ejpam-546	208	5	gardner	gardner	NOUN
ejpam-546	208	6	.	.	PUNCT
ejpam-546	209	1	mathematical	mathematical	ADJ
ejpam-546	209	2	games	games	PROPN
ejpam-546	209	3	.	.	PUNCT
ejpam-546	210	1	scientific	scientific	ADJ
ejpam-546	210	2	american	american	PROPN
ejpam-546	210	3	,	,	PUNCT
ejpam-546	210	4	page	page	NOUN
ejpam-546	210	5	115	115	NUM
ejpam-546	210	6	,	,	PUNCT
ejpam-546	210	7	october	october	PROPN
ejpam-546	210	8	1973	1973	NUM
ejpam-546	210	9	.	.	PUNCT
ejpam-546	211	1	[	[	X
ejpam-546	211	2	5	5	NUM
ejpam-546	211	3	]	]	X
ejpam-546	211	4	s	s	PART
ejpam-546	211	5	kung	kung	PROPN
ejpam-546	211	6	.	.	PROPN
ejpam-546	211	7	sum	sum	PROPN
ejpam-546	211	8	of	of	ADP
ejpam-546	211	9	squares	square	NOUN
ejpam-546	211	10	.	.	PUNCT
ejpam-546	212	1	college	college	NOUN
ejpam-546	212	2	mathematics	mathematics	PROPN
ejpam-546	212	3	journal	journal	PROPN
ejpam-546	212	4	,	,	PUNCT
ejpam-546	212	5	(	(	PUNCT
ejpam-546	212	6	20):205	20):205	NUM
ejpam-546	212	7	,	,	PUNCT
ejpam-546	212	8	1989	1989	NUM
ejpam-546	212	9	.	.	PUNCT
ejpam-546	213	1	[	[	X
ejpam-546	213	2	6	6	NUM
ejpam-546	213	3	]	]	PUNCT
ejpam-546	213	4	l	l	PROPN
ejpam-546	213	5	larson	larson	PROPN
ejpam-546	213	6	.	.	PUNCT
ejpam-546	214	1	a	a	DET
ejpam-546	214	2	discrete	discrete	ADJ
ejpam-546	214	3	look	look	NOUN
ejpam-546	214	4	at	at	ADP
ejpam-546	214	5	1	1	NUM
ejpam-546	214	6	+	+	SYM
ejpam-546	214	7	2	2	NUM
ejpam-546	214	8	+	+	NUM
ejpam-546	214	9	.	.	PUNCT
ejpam-546	214	10	.	.	PUNCT
ejpam-546	215	1	.+	.+	PROPN
ejpam-546	215	2	n.	n.	PROPN
ejpam-546	215	3	college	college	PROPN
ejpam-546	215	4	mathematics	mathematics	PROPN
ejpam-546	215	5	journal	journal	NOUN
ejpam-546	215	6	,	,	PUNCT
ejpam-546	215	7	(	(	PUNCT
ejpam-546	215	8	16):369–382	16):369–382	NOUN
ejpam-546	215	9	,	,	PUNCT
ejpam-546	215	10	1985	1985	NUM
ejpam-546	215	11	.	.	PUNCT
ejpam-546	216	1	[	[	X
ejpam-546	216	2	7	7	X
ejpam-546	216	3	]	]	PUNCT
ejpam-546	216	4	w	w	NOUN
ejpam-546	216	5	lushbaugh	lushbaugh	PROPN
ejpam-546	216	6	.	.	PUNCT
ejpam-546	217	1	(	(	PUNCT
ejpam-546	217	2	no	no	DET
ejpam-546	217	3	title	title	NOUN
ejpam-546	217	4	)	)	PUNCT
ejpam-546	217	5	.	.	PUNCT
ejpam-546	218	1	mathematical	mathematical	ADJ
ejpam-546	218	2	gazette	gazette	PROPN
ejpam-546	218	3	,	,	PUNCT
ejpam-546	218	4	(	(	PUNCT
ejpam-546	218	5	49):200	49):200	PROPN
ejpam-546	218	6	,	,	PUNCT
ejpam-546	218	7	1965	1965	NUM
ejpam-546	218	8	.	.	PUNCT
ejpam-546	219	1	[	[	X
ejpam-546	219	2	8	8	NUM
ejpam-546	219	3	]	]	X
ejpam-546	219	4	r	r	NOUN
ejpam-546	219	5	nelsen	nelsen	NOUN
ejpam-546	219	6	.	.	PUNCT
ejpam-546	220	1	proofs	proof	NOUN
ejpam-546	220	2	without	without	ADP
ejpam-546	220	3	words	word	NOUN
ejpam-546	220	4	:	:	PUNCT
ejpam-546	220	5	exercises	exercise	NOUN
ejpam-546	220	6	in	in	ADP
ejpam-546	220	7	visual	visual	ADJ
ejpam-546	220	8	thinking	thinking	NOUN
ejpam-546	220	9	.	.	PUNCT
ejpam-546	221	1	mathematical	mathematical	ADJ
ejpam-546	221	2	association	association	PROPN
ejpam-546	221	3	of	of	ADP
ejpam-546	221	4	america	america	PROPN
ejpam-546	221	5	,	,	PUNCT
ejpam-546	221	6	washington	washington	PROPN
ejpam-546	221	7	,	,	PUNCT
ejpam-546	221	8	1993	1993	NUM
ejpam-546	221	9	.	.	PUNCT
ejpam-546	222	1	[	[	X
ejpam-546	222	2	9	9	NUM
ejpam-546	222	3	]	]	X
ejpam-546	222	4	r	r	NOUN
ejpam-546	222	5	nelsen	nelsen	NOUN
ejpam-546	222	6	.	.	PUNCT
ejpam-546	223	1	proofs	proof	NOUN
ejpam-546	223	2	without	without	ADP
ejpam-546	223	3	words	word	NOUN
ejpam-546	223	4	ii	ii	PROPN
ejpam-546	223	5	:	:	PUNCT
ejpam-546	223	6	more	more	ADJ
ejpam-546	223	7	exercises	exercise	NOUN
ejpam-546	223	8	in	in	ADP
ejpam-546	223	9	visual	visual	ADJ
ejpam-546	223	10	thinking	thinking	NOUN
ejpam-546	223	11	.	.	PUNCT
ejpam-546	224	1	mathematical	mathematical	ADJ
ejpam-546	224	2	association	association	PROPN
ejpam-546	224	3	of	of	ADP
ejpam-546	224	4	america	america	PROPN
ejpam-546	224	5	,	,	PUNCT
ejpam-546	224	6	washington	washington	PROPN
ejpam-546	224	7	,	,	PUNCT
ejpam-546	224	8	2000	2000	NUM
ejpam-546	224	9	.	.	PUNCT
ejpam-546	225	1	[	[	X
ejpam-546	225	2	10	10	NUM
ejpam-546	225	3	]	]	X
ejpam-546	225	4	f	f	PROPN
ejpam-546	225	5	pouryoussefi	pouryoussefi	PROPN
ejpam-546	225	6	.	.	PUNCT
ejpam-546	226	1	proof	proof	NOUN
ejpam-546	226	2	without	without	ADP
ejpam-546	226	3	words	word	NOUN
ejpam-546	226	4	.	.	PUNCT
ejpam-546	227	1	mathematics	mathematic	NOUN
ejpam-546	227	2	magazine	magazine	NOUN
ejpam-546	227	3	,	,	PUNCT
ejpam-546	227	4	(	(	PUNCT
ejpam-546	227	5	62):323	62):323	ADJ
ejpam-546	227	6	,	,	PUNCT
ejpam-546	227	7	1989	1989	NUM
ejpam-546	227	8	.	.	PUNCT
ejpam-546	228	1	[	[	X
ejpam-546	228	2	11	11	NUM
ejpam-546	228	3	]	]	X
ejpam-546	228	4	s	s	PART
ejpam-546	228	5	stein	stein	PROPN
ejpam-546	228	6	.	.	PUNCT
ejpam-546	229	1	existence	existence	NOUN
ejpam-546	229	2	out	out	ADP
ejpam-546	229	3	of	of	ADP
ejpam-546	229	4	chaos	chaos	NOUN
ejpam-546	229	5	.	.	PUNCT
ejpam-546	230	1	in	in	ADP
ejpam-546	230	2	r	r	PROPN
ejpam-546	230	3	honsberger	honsberger	NOUN
ejpam-546	230	4	,	,	PUNCT
ejpam-546	230	5	editor	editor	NOUN
ejpam-546	230	6	,	,	PUNCT
ejpam-546	230	7	mathematical	mathematical	ADJ
ejpam-546	230	8	plums	plum	NOUN
ejpam-546	230	9	,	,	PUNCT
ejpam-546	230	10	pages	page	NOUN
ejpam-546	230	11	62–93	62–93	NUM
ejpam-546	230	12	.	.	PUNCT
ejpam-546	231	1	mathematical	mathematical	PROPN
ejpam-546	231	2	association	association	PROPN
ejpam-546	231	3	of	of	ADP
ejpam-546	231	4	america	america	PROPN
ejpam-546	231	5	,	,	PUNCT
ejpam-546	231	6	washington	washington	PROPN
ejpam-546	231	7	,	,	PUNCT
ejpam-546	231	8	1979	1979	NUM
ejpam-546	231	9	.	.	PUNCT