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?

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

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 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.

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

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.

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

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?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

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

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

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

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?

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

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

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

Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?

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?

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

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?

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.

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.

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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.

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.

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?

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?

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

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?

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?

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

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

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

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

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

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Charlie has moved between countries and the average income of both has increased. How can this be so?

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

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?