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

Can you explain the strategy for winning this game with any target?

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

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.

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

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

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

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?

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?

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

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?

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?

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?

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

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?

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?

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?

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

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

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?

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

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.

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

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

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

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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

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

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?

An investigation that gives you the opportunity to make and justify predictions.

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.

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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?

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.

Are these statements always true, sometimes true or never true?

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

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

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

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

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.

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.