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Square Triangles

I often read, and use, Emily Hughes’s (@ilovemathsgames) “Puzzles of the week“. This week’s included this:

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I liked the puzzle, it’s fun and it fits well with the sequence of lessons my year 9s have completed recently, but it’s basically a trial and error question. There are 13 square numbers below 180, and it wouldn’t take too long to plug each one it, especially with a computer.

I set myself the challenge of having only one guess, and started to think strategically. Firstly, it’s impossible for the solution to contain an odd number of odd numbers. Ie, there’s 1 even number in the solution or all the numbers are even. Then I thought about making ten with the final digits of the squares. The most obvious seemed to be 0,4,6. Which would mean 100, 64 and 16, which is a solution.

I wondered if there were more, but I’ve considered the problem and I’m pretty sure there isn’t. I am now thinking if something similar can be done for other polygons, can 360 be expressed as the sum of 4 squares? Can 540 be expressed as the sum of 5? Food for thought.

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