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?

What is the greatest number of squares you can make by overlapping three squares?

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

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.

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

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

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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!

A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.

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

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

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?

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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

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?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

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

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?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

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

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....?

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

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?

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 shaped overlaps can you make with two circles which are the same size? What shapes are 'left over'? What shapes can you make when the circles are different sizes?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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?

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. . . .

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

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

A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?

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

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

A game in which players take it in turns to choose a number. Can you block your opponent?

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

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?

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

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.

Try out the lottery that is played in a far-away land. What is the chance of winning?

Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?

Can you explain the strategy for winning this game with any target?

The 2012 primary advent calendar features twenty-four of our posters, one for each day in the run-up to Christmas.