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

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 explain the strategy for winning this game with any target?

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

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

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?

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?

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.

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

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

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 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.

Delight your friends with this cunning trick! Can you explain how it works?

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.

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

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

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

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

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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

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?

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.

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'.

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

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

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.

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

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

What happens when you round these three-digit numbers to the nearest 100?

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

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?

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

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

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?

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?

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?

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

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?

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.

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

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

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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

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.