Triangle numbers

Can you find any two-digit numbers that satisfy all of these statements?

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

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?

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

Can you find a rule which connects consecutive triangular numbers?

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

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

These alphabet bricks are painted in a special way. A is on one brick, B on two bricks, and so on. How many bricks will be painted by the time they have got to other letters of the alphabet?

Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.