Logic puzzles, riddles, & maths problems

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Please help me identify these 100 light bulbs by turning ON and OFF their switches.

This problem is a step up from Problem #028 - hidden key. Can you tackle this one?

There is a key hidden in one of three boxes and each box has a coin on top of it. Can you use the coins to let your friend know where the key is hiding?

Some people are standing quiet in a line, each person with a hat that has one of two colours. How many people can guess their colour correctly?

This problem is a really interesting problem I solved two times. The first time I solved it I failed to prove exactly how it works... then some years later I remembered the problem statement and was able to solve it properly. Let's see how you do!