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Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

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Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Triangular Triples

Stage: 3 Challenge Level: Challenge Level:1

Three numbers a, b and c are a Pythagorean triple if $a^2+ b^2= c^2$. The triangular numbers are:

$\frac{1\times 2}{2}, \frac{2\times 3}{2}, \frac{3\times 4}{2}, \frac{4\times 5}{2}$

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple. [In fact, these are the ONLY known set of three triangular numbers that form a Pythagorean triple.]