### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### Happy Numbers

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.

### Intersecting Circles

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:

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.]