A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

Here are shadows of some 3D shapes. What shapes could have made them?

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?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

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

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

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

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

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?

Explore the lattice and vector structure of this crystal.

Use the diagram to investigate the classical Pythagorean means.

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

Well done to all of you who found both solutions to this problem. We received some very well-explained solutions.

We received a variety of well explained solutions to this problem, starting with the specific and finishing with the general. Thank you all.

Rajeev and Christian both explained their thinking about this probability problem very clearly.

What would you do if your teacher asked you add all the numbers from 1 to 100? Find out how Carl Gauss responded when he was asked to do just that.

In this article, we look at solids constructed using symmetries of their faces.

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

An article introducing continued fractions with some simple puzzles for the reader.

A java applet that takes you through the steps needed to solve a Diophantine equation of the form Px+Qy=1 using Euclid's algorithm.