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

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.

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?

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

A game for 2 players with similaritlies 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 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.

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

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.

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

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

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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?

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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.

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

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

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

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?

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?

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

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

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.

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.

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.

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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

Can you find all the ways to get 15 at the top of this triangle of numbers?

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

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

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.

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

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

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

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

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

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?

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

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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

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?

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?

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

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

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

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

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