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

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

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

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?

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.

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

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?

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

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

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

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?

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

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

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

This task follows on from Build it Up and takes the ideas into three dimensions!

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

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

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

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?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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.

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

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. . . .

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

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

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

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

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?

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

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

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.

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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.

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

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

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?

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

Can you find all the ways to get 15 at the top of this triangle of numbers?

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?

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?

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