In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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

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

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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

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 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.

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

This challenge asks you to imagine a snake coiling on itself.

Find out what a "fault-free" rectangle is and try to make some of your own.

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

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

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

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

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Here are two kinds of spirals for you to explore. What do you notice?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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

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.

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

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

An investigation that gives you the opportunity to make and justify predictions.

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

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

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

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 winner is the player to take the last counter.

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?

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

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

Find the sum of all three-digit numbers each of whose digits is odd.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

What happens when you round these three-digit numbers to the nearest 100?

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?