Can you find sets of sloping lines that enclose a square?

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?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

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

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.

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

Can all unit fractions be written as the sum of two unit fractions?

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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.

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.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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?

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

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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?”

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.

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.

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.

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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

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.

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.

How many moves does it take to swap over some red and blue frogs? Do you have a method?

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?

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

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

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

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?

Can you explain the strategy for winning this game with any target?

It starts quite simple but great opportunities for number discoveries and patterns!

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

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

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

Got It game for an adult and child. How can you play so that you know you will always win?

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

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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

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?

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?

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