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Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Triangular Triples

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

Iff

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

Square Sum

Age 14 to 16 Short
Challenge Level

Answer: 148

    110 + 111 + 112 + 113 + 114 + 115 + 116 + 117 + 118
or 120 + ... + 128
     etc
sum = 9$\times$ middle number
114$\times$9
124$\times$9
134$\times$9
144$\times$9 = 12$^2\times$3$^2$ = (12$\times$3)$^2$ so is a square number

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.