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?

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?

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?

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