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

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

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

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

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.

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

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

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

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

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

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

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?

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 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.

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?

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

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 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?

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

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

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

Watch the video of Fran re-ordering these number cards. What do you notice? Try it for yourself. 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.

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

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

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

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

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

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

What happens when you round these numbers to the nearest whole number?

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?

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

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

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

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

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

Watch this animation. What do you see? Can you explain why this happens?

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!

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

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?

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.

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

Can you find a way of counting the spheres in these arrangements?

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.