Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

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

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

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

Help the bee to build a stack of blocks far enough to save his friend trapped in the tower.

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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?

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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

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

Practise your diamond mining skills and your x,y coordination in this homage to Pacman.

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?

A game for 2 people that everybody knows. You can play with a friend or online. If you play correctly you never lose!

A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.

A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

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!

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

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?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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?

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a 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?

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?

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

NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.

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....?

Can you beat the computer in the challenging strategy game?

Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.

Exchange the positions of the two sets of counters in the least possible number of moves

Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?

Train game for an adult and child. Who will be the first to make the train?

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.

Work out the fractions to match the cards with the same amount of money.

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

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?

Can you complete this jigsaw of the multiplication square?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

Square It game for an adult and child. Can you come up with a way of always winning this game?

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

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .