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

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

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

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

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.

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?

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

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

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

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

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.

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

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.

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

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

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

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

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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

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?

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

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

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

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?

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

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

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.

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.

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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

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

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

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 find sets of sloping lines that enclose a square?

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.

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.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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.

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

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

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

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .