These alphabet bricks are painted in a special way. A is on one brick, B on two bricks, and so on. How many bricks will be painted by the time they have got to other letters of the alphabet?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Where can you draw a line on a clock face so that the numbers on both sides have the same total?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

A church hymn book contains 700 hymns. The numbers of the hymns are displayed by combining special small single-digit boards. What is the minimum number of small boards that is needed?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

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

Which set of numbers that add to 10 have the largest product?

Is there an efficient way to work out how many factors a large number has?

We received many detailed solutions to this challenge, most of which made use of a spreadsheet.

The Four Colour Conjecture was first stated just over 150 years ago, and finally proved conclusively in 1976. It is an outstanding example of how old ideas can be combined with new discoveries. prove a mathematical theorem.