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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?
Can you work out what step size to take to ensure you visit all the dots on the circle?
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?
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?
Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?
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.
If you move the tiles around, can you make squares with different coloured edges?
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
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?
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?
A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?
How many different symmetrical shapes can you make by shading triangles or squares?
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.
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
In how many ways can you fit all three pieces together to make shapes with line symmetry?
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
Some diagrammatic 'proofs' of algebraic identities and inequalities.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?
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?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
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?
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
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 metres from B. How long is the lake?
Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
Is there an efficient way to work out how many factors a large number has?
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.