How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

In your bank, you have three types of coins. The number of spots shows how much they are worth. Can you choose coins to exchange with the groups given to make the same total?

Starting with the number 180, take away 9 again and again, joining up the dots as you go. Watch out - don't join all the dots!

Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?

Make one big triangle so the numbers that touch on the small triangles add to 10.

Use the information about Sally and her brother to find out how many children there are in the Brown family.

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Investigate which numbers make these lights come on. What is the smallest number you can find that lights up all the lights?

Here is a chance to play a version of the classic Countdown Game.

Move just three of the circles so that the triangle faces in the opposite direction.

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

There are three versions of this challenge. The idea is to change the colour of all the spots on the grid. Can you do it in fewer throws of the dice?

Mr Gilderdale is playing a game with his class. What rule might he have chosen? How would you test your idea?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Can you hang weights in the right place to make the equaliser balance?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Can you make the green spot travel through the tube by moving the yellow spot? Could you draw a tube that both spots would follow?

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

Can you use the numbers on the dice to reach your end of the number line before your partner beats you?

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

If you have only four weights, where could you place them in order to balance this equaliser?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Twenty four games for the run-up to Christmas.

How many trains can you make which are the same length as Matt's, using rods that are identical?

Use the interactivity to find out how many quarter turns the man must rotate through to look like each of the pictures.

Can you find all the different triangles on these peg boards, and find their angles?

Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.

How many different triangles can you make on a circular pegboard that has nine pegs?

Can you find all the different ways of lining up these Cuisenaire rods?

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

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

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

Yasmin and Zach have some bears to share. Which numbers of bears can they share so that there are none left over?

How many right angles can you make using two sticks?

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

Ahmed has some wooden planks to use for three sides of a rabbit run against the shed. What quadrilaterals would he be able to make with the planks of different lengths?