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?

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?

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

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.

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

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.

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?

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

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

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

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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?

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

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.

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.

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

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.

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

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?

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

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.

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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.

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?

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

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?

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

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

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.

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

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

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.

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

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

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

Charlie has moved between countries and the average income of both has increased. How can this be so?

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

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

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

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

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?

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?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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

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