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What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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An investigation that gives you the opportunity to make and justify predictions.

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Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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This task follows on from Build it Up and takes the ideas into three dimensions!

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Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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What happens when you round these three-digit numbers to the nearest 100?

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While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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How many centimetres of rope will I need to make another mat just like the one I have here?

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How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

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Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

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Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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What happens when you round these numbers to the nearest whole number?

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In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

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Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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Try out this number trick. What happens with different starting numbers? What do you notice?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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Got It game for an adult and child. How can you play so that you know you will always win?

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Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

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Find the sum of all three-digit numbers each of whose digits is odd.

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Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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Surprise your friends with this magic square trick.

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Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

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In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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Delight your friends with this cunning trick! Can you explain how it works?