Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
What's the largest volume of box you can make from a square of paper?
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?
To avoid losing think of another very well known game where the patterns of play are similar.
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?
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.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
Can you describe this route to infinity? Where will the arrows take you next?
A game for 2 players
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?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
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
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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?
Can you use the diagram to prove the AM-GM inequality?
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
Can you find the values at the vertices when you know the values on the edges?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
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.
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. . . .
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
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. . . .
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
This task encourages you to investigate the number of edging pieces and panes in different sized windows.