For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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.

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.

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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.

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?

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

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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

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.

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

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.

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

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

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.

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

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

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 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?

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. . . .

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

To avoid losing think of another very well known game where the patterns of play are similar.

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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.

Explore the effect of combining enlargements.

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 find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

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.

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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.

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 loser is the player who takes the last counter.

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

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

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

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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

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?