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?

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

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

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?

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

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

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

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.

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.

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

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?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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

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.

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

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?

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

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.

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

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.

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

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.

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

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

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

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.

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.

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

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?

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.

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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?

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

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.

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

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

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

Can you explain the strategy for winning this game with any target?

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. . . .

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

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

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

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?

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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?

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.