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.

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

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

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

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

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

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

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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

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.

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

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.

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

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

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.

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.

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

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

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

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

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?

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

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

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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

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?

This task follows on from Build it Up and takes the ideas into three dimensions!

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?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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

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?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. 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.

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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?

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?

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

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

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?

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 make dice stairs using the rules stated? How do you know you have all the possible stairs?

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?