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

The clues for this Sudoku are the product of the numbers in adjacent squares.

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?

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

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

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?

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

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.