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?

Sam displays cans in 3 triangular stacks. With the same number he could make one large triangular stack or stack them all in a square based pyramid. How many cans are there how were they arranged?

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Watch the video to see how Charlie works out the sum. Can you adapt his method?

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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

Here is a collection of puzzles about Sam's shop sent in by club members. Perhaps you can make up more puzzles, find formulas or find general methods.

Can you describe this route to infinity? Where will the arrows take you next?

Prove that if n is a triangular number then 8n+1 is a square number. Prove, conversely, that if 8n+1 is a square number then n is a triangular number.

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

I have forgotten the number of the combination of the lock on my briefcase. I did have a method for remembering it...