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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

Well explained by Tom, from Cottenham Village College

The answer I got for $T_n$ is: $8T_n+1=(2n+1)^2$

The $2n+1$ part is because the diagram looks like this for $T_3$
square arrangement
$2T_n$ form a rectangle $n$ by $n+1$

The four rectangles rotate around the centre and together make a square of side $n+(n+1)$

So we get the equation $8T_n+1=(2n+1)^2$