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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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My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

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Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

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

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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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In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

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

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Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

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Can you find a rule which connects consecutive triangular numbers?

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This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I 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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Can you see how to build a harmonic triangle? Can you work out the next two rows?

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The sums of the squares of three related numbers is also a perfect square - can you explain why?

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

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

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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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A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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Can you find a rule which relates triangular numbers to square numbers?

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Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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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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If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

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Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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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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Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

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Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

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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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Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

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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?

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Show that all pentagonal numbers are one third of a triangular number.

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

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

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The Number Jumbler can always work out your chosen symbol. Can you work out how?

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

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Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

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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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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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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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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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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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To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

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Where should you start, if you want to finish back where you started?