The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Where should you start, if you want to finish back where you started?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

Play around with sets of five numbers and see what you can discover about different types of average...

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

How many different symmetrical shapes can you make by shading triangles or squares?

If you move the tiles around, can you make squares with different coloured edges?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

The clues for this Sudoku are the product of the numbers in adjacent squares.

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Six balls of various colours are randomly shaken into a trianglular arrangement. What is the probability of having at least one red in the corner?

What happens when you add a three digit number to its reverse?

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Can you find a way to identify times tables after they have been shifted up?

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

Engage in a little mathematical detective work to see if you can spot the fakes.

Find the frequency distribution for ordinary English, and use it to help you crack the code.

Can you do a little mathematical detective work to figure out which number has been wiped out?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

What is the smallest number of answers you need to reveal in order to work out the missing headers?

Can you find the values at the vertices when you know the values on the edges?

Use the differences to find the solution to this Sudoku.

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

There are nasty versions of this dice game but we'll start with the nice ones...

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...

Who said that adding, subtracting, multiplying and dividing couldn't be fun?