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?
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
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?
Can you describe this route to infinity? Where will the arrows take you next?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
A game for 2 players
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.
If you move the tiles around, can you make squares with different coloured edges?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Can you maximise the area available to a grazing goat?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
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. . . .
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Can you discover whether this is a fair game?
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?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?
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 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?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?
To avoid losing think of another very well known game where the patterns of play are similar.
Show that all pentagonal numbers are one third of a triangular number.
Can you find a rule which connects consecutive triangular numbers?
Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
How many different symmetrical shapes can you make by shading triangles or squares?
Show that among the interior angles of a convex polygon there cannot be more than three acute angles.
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?
An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
A huge wheel is rolling past your window. What do you see?
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
Can you find a rule which relates triangular numbers to square numbers?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.
Can you use the diagram to prove the AM-GM inequality?
You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.