Triangle numbers

Take ten sticks in heaps any way you like. Make a new heap using one from each of the heaps. By repeating that process could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?

Show that all pentagonal numbers are one third of a triangular number.

Can you find a rule which relates triangular numbers to square numbers?

Can you find a rule which connects consecutive triangular numbers?

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.

Does a graph of the triangular numbers cross a graph of the six times table? If so, where? Will a graph of the square numbers cross the times table too?

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

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

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...