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For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

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Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

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A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

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A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

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A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

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Can you find the link between these beautiful circle patterns and Farey Sequences?

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A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

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See if you can anticipate successive 'generations' of the two animals shown here.

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A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

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A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

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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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A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

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The net of a cube is to be cut from a sheet of card 100 cm square. What is the maximum volume cube that can be made from a single piece of card?

Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?

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The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

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Use the diagram to investigate the classical Pythagorean means.

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

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The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

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Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

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A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

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Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

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We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

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The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

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

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In this problem we see how many pieces we can cut a cube of cheese into using a limited number of slices. How many pieces will you be able to make?

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Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .

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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.

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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.

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A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

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A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

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Can you find a rule which connects consecutive triangular numbers?

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There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

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

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

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Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

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A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.

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A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

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This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?

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Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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What's the largest volume of box you can make from a square of paper?

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What can you see? What do you notice? What questions can you ask?

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A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

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Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?