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

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Can you describe this route to infinity? Where will the arrows take you next?

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

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.

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

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?

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

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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?

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

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?

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?

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?

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

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

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.

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

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

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.

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.

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

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?

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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.

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

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

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

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.

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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

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

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

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

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?

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?

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up 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.

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.

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.

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. . . .