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Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

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What is the total number of squares that can be made on a 5 by 5 geoboard?

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A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

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Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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Surprising numerical patterns can be explained using algebra and diagrams...

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Can you see how to build a harmonic triangle? Can you work out the next two rows?

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When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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Can you find the area of a parallelogram defined by two vectors?

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Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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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?

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Make some loops out of regular hexagons. What rules can you discover?

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First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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How good are you at finding the formula for a number pattern ?

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Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

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The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so. . . .

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How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

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The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

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Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

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Can you explain what is going on in these puzzling number tricks?

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

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Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

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The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

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What is special about the difference between squares of numbers adjacent to multiples of three?

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Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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There are unexpected discoveries to be made about square numbers...

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Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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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?

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Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

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A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

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Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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Can you make sense of these three proofs of Pythagoras' Theorem?

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A task which depends on members of the group noticing the needs of others and responding.

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Play around with the Fibonacci sequence and discover some surprising results!

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Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?