# Puzzle: Figuring Out Figurate Numbers

Photo by frumbert.

Alexandria Jones‘s parents decided that the family needed to relax after the excitement of tracking Simon Skulk, so they spent the next day at a beach on the Mediterranean coast. Leon collected pebbles and tried to build up figurate numbers — numbers that make a figure, or shape — the way Dr. Theano had shown them.

## Triangular Numbers

He started with one pebble, since all the figurate numbers grow from the number one. Two more pebbles made a triangle, the number 3. He couldn’t make a triangle with 4 or 5, so the next triangular number was 6.

As Leon added pebbles, he kept a list of the triangular numbers: 1, 3, 6, 10, 15… Then he called Alex over to look at the pattern he found.

• Can you find the pattern in the triangular numbers?
• If n stands for any number, can you write a formula to calculate the nth triangular number?

## Square Numbers

Alex had studied square numbers before, but she had never thought of them as “numbers that make a square.” She helped Leon build the square numbers, and they kept a list: 1, 4, 9, 16, 25…

“Hey,” she said, “these numbers make a cool pattern, too. And look, each square number is the sum of two triangular numbers!”

• Can you find the pattern Alex noticed?
• Can you show that every square number is the sum of two triangular numbers?
• What 2-digit number is both square and triangular?
• If n stands for any number, can you write a formula for the nth square number?

## Pentagonal Numbers

They found the pentagonal numbers more confusing than the earlier figures. Leon drew lines on the sand to mark the corners of each pentagon, which helped. Beginning with one (of course!), the pentagonal numbers were: 1, 5, 12, 22, 35…

“That pattern isn’t as interesting as the other two,” he said. “But this is neat: each pentagon is the sum of a square and a triangle.”

• How is that true?
• If n stands for any number, can you write the formula for the nth pentagonal number?
• Can you find a 3-digit number that is both triangular and pentagonal? How about a 4-digit number that is both square and pentagonal?
• Can you find a number that is triangular AND square AND pentagonal?

Think you have these puzzles figured out? Check your solutions here:

## To Be Continued…

Read all the posts from the March/April 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.

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## 11 thoughts on “Puzzle: Figuring Out Figurate Numbers”

1. How comes I never got a chance to learn about this site before? Denise, you’ve indeed managed to build a wealth of web-based math resources I can only dream of. I particularly love the kind of mind-storm these exercises generate. Thank you for that.

Now, I guess it’s high time for me to revive my blog 😉

Keep up the faith,
Jean-Marie

2. I am glad you are enjoying my puzzles. Watch for a related strategy game coming later this week…

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5. amyswandering says:

This is fascinating – thanks for sharing!

6. Pingback: Are we there yet?
7. mark says:

Extraterrestrial Intelligence
BY HOWARD H. CAMPAIGNE
Unclassified
,
Extraterrestrial Communications
In the most recent issue of theNSA Technical Joumal- Vol. XI, No. 1-
Mr. Lambros D. Callimahos discussed certain aspects of extraterrestrial
intelligence and included seueral messages to test the reader’s ingenuity.
In the foUowing pages, Dr. H. H. Campaigne offers additional commu-
nications /rom outer space.
Recently a series of radio messages was heard coming from outer
space. The transmission was not continuous but was cut by pauses
into pieces which could be taken as units, for they were repeated over
and over again. The pauses show here as punctuation. The various
combinations have been represented by letters of the alphabet, so that
the messages can be written down. Each message except the first is
given here only once. The serial number of the message has been
supplied for each reference.
1 . ABCDEFGHJKLMNOPQRSTUV
ABCDEFCHJKLMNOPQRSTUV
ABCDEFGHJKLMNOPQR etc.
2. AA. B;AAA.C; AAAA.D; AAAAA.E; AAAAAA.F; AAAAAAA.G.
3. LAA; LBB. LCC; LDD; LEE; LFF; LGG.
4. LBKAA; LCKBA; LCKAB; LDKCA; LOKBB; LOKAC; LEKDA;
LEKCB; LEKBC; LEKAD: LFKEA; LFKDB; LFKCC; LFKBD;
LFKAE.
5. LFKAKBC; LFKCKBA; LGKAKBD; LGKCKAC; LKAKBCKKABC.
6. LAllBA; LBYCA; LAMCB; LCllDA; LBIIDB; LAMDC; LDMEA;
LCOB; LBMEC; LAMED.
7. LNIIAA; LNYBB; LNMCC; LNMDD; LNIIEE; LNMFF; LNMGG.
8. LAOAA; LNONA; LBOAB; LBOBA; LNONB; LNOBN; LOOBB;
9. LFOAOBC; LFOCOBA; LFOBOCA; LODOEFOODEF.
10. LDRBB; LBRBA; LARBH; LCRCA; LARCN.
11 . LRBCKDD; LRBCKAG; LRBCOBD; LRBBD.
12. LRCBKDE; LRCBOCC.
13. WRBC; WKAG; LKJAOCC; LKJARCB.
14. LBPJD; LDPJB; LAPCC; LCPFB.

8. Wow! I’ve never been patient enough to figure out codes, but I bet someone will have fun with this.

9. 11-22-08

Aloha,

Thanks for the insight of the possible 1st school of Pythagoras, is there any new insight?

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