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

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

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

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?

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

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 explain the strategy for winning this game with any target?

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.

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

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.

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

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

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

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.

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.

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

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

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

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

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

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

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

This challenge asks you to imagine a snake coiling on itself.

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.

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

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

Surprise your friends with this magic square trick.

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

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

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.

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

Can you find the values at the vertices when you know the values on the edges?

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

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

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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

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

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

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

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

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.

Can all unit fractions be written as the sum of two unit fractions?

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

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?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?