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

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

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?

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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