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?

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

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

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

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

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?

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

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

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?

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

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

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

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.

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

This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

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

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

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.

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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

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

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

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

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

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

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?

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

This challenge is about finding the difference between numbers which have the same tens digit.

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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

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?

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?

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?

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?

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.

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?

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?

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

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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.

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.

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?

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?