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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.


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

Triangles Within Squares

Age 14 to 16 Challenge Level:
There is a very strong connection between this problem and the "Sequences and Series" problem.

Can you see the rectangles made from two triangular numbers in the square?

Can you explain why there is always one left over?