Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

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.

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

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.

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.

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

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

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

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

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

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?

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

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.

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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

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.

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

Can you find sets of sloping lines that enclose a square?

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?

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?

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

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

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

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.

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

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. . . .

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

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?

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

Can you describe this route to infinity? Where will the arrows take you next?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

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 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?

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.

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.

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

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.

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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

It would be nice to have a strategy for disentangling any tangled ropes...

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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