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?

Answers

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

• Answers to Leon’s Figurate Number Puzzles

To Be Continued…

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

• Alexandria Jones and the Puzzling Pythagorean Pebbles