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

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 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 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.

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

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.

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.

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

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.

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

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.

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?

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.

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

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

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

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

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

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.

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

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

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

These tasks give learners chance to generalise, which involves identifying an underlying structure.

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

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.

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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?

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

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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?

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

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

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

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?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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

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.

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

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

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

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?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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

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

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