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?

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?

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

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

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

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. . . .

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.

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

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

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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

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

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?

Make some loops out of regular hexagons. What rules can you discover?

What's the largest volume of box you can make from a square of paper?

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

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

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

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.

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.

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?

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

Explore the effect of reflecting in two intersecting mirror lines.

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

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

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

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

Explore the effect of combining enlargements.

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

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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?

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?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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. . . .

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

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

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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

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. . . .

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

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.

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

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

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

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