GeekDad Puzzle of the Week Solution – Right Triangle Numbers

Reading Time: 2 minutes

This past week’s puzzle:

triangle_story

This past weekend, as Max was learning to stack cups quickly, he and I discussed triangle numbers. One way to describe the set of triangle numbers is the set of numbers that make complete rows when stacking cups or laying out bowling pins: 1, 3, 6, 10, 15, 21, etc.

The first 6 triangle numbers (Source: wikipedia)
The first 6 triangle numbers
(Source: wikipedia)

Pairs of consecutive triangle numbers have unusual properties: not only is the sum of any two consecutive triangle numbers always square, but it is also the case that the sum of the squares of two consecutive triangle numbers is always another triangle number! (Go ahead, try a few out… I’ll wait.)

So, do triangle numbers have anything to do with right triangles? That’s the crux of this week’s GeekDad Puzzle of the Week.

For your chance at this week’s fabulous prize answer this question: Are there any sets of three triangle numbers that could be the sides of a right triangle? If not, why not?

Congratulations to this week’s winner, Blaine Felicia. Blaine correctly located the one known trio of triangle numbers that can make the sides of a right triangle:

T(132)² + T(143)² = T(164)²
8778² + 10296² = 13530²
77053284 + 106007616 = 183060900
183060900 = 183060900

Many thanks to everyone that submitted an answer to this past week’s puzzle. For being drawn at random from all of the correct submissions, a $50 Gift Certificate from ThinkGeek will be on its way to Blaine shortly.

Happy puzzling!

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