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?

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.

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

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

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?

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

Stop the Clock game for an adult and child. How can you make sure you always win this game?

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

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

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

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

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

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

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

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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

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

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

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

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

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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.

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

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

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?

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

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 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?

This activity involves rounding four-digit numbers to the nearest thousand.

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

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?

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 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.

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?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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?

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.

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

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

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.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?