Rotating a pencil twice about two different points gives surprising results...

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Ben’s class were making cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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

Think of a number... follow the machine's instructions. I know what your number is! Can you explain how I know?

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

A combination of logical thinking and trial and improvement helped Nicky, Bronya, Leila and Sheya to solve this problem.

Three different approaches explain the solution to this problem. Particular thanks to Michael, Peter and Ian.

David has given a neat solution and also written a program to calculate big powers in modulus arithmetic.

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

A shunting puzzle for 1 person. Swop the positions of the counters at the top and bottom of the board.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Can you beat Piggy in this simple dice game? Can you figure out Piggy's strategy, and is there a better one?