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?

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?

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

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

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

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?

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

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

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

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

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

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?

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

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

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

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

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

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?

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

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

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

Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning?

Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?

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

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?

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

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

Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses?

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

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

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

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. . . .

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?

Use the differences to find the solution to this Sudoku.

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

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

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

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?

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

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 try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?

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

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...

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

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?

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

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

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