Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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.

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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.

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. . . .

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?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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?

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.

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.

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

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 explain the surprising results Jo found when she calculated the difference between square numbers?

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

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?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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.

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Can you find the values at the vertices when you know the values on the edges?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

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?

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

Charlie has moved between countries and the average income of both has increased. How can this be so?

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

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 winner is the player to take the last counter.

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?

An account of some magic squares and their properties and and how to construct them for yourself.

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

To avoid losing think of another very well known game where the patterns of play are similar.

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

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...