Can you crack these very difficult challenge ciphers? How might you systematise the cracking of unknown ciphers?

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Can you set the logic gates so that this machine can decide how many bulbs have been switched on?

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?