Puzzle 5: Save the prisoners

There are 100 prisoners. The warden will make them stand in a queue and put black or white hats on each of them. Each prisoner can see the color of the hats of prisoners ahead of them only, and not theirs or the ones behind. If any prisoner can correctly say the color of their hat, he/she will be freed, else hanged. They can guess in any order. The prisoners talk beforehand to come up with a strategy to save the most prisoners possible. What is the most number of prisoners they can save?

Puzzle 4: White and black balls

There are 100 white and 100 black balls in a bin. You take out two balls at a time. If both are the same colour you replace them with a black ball. If they are different, you replace them with a white one. When you are down to one ball, what is the colour of that one ball? (No answers on comments please :))

Puzzle 3: Two buses and a bee

Bus 1 is leaving San Jose for LA, and bus 2 is leaving LA for San Jose at the same time. Bus 1 travels at 50 miles an hour and Bus 2 at 35 miles an hour. A bee sitting on Bus 1 at the start keeps flying from Bus 1 to Bus 2 to Bus 1 etc. at 70 miles an hour. If LA and San Jose are 500 miles apart, how far will the bee have flown when the 2 buses cross each other?

Puzzle 2: Handshakes in the world

Of all the people in the world, prove that the number of people who have shaken hands an odd number of times, is even

Puzzle 1: Handshakes in a room

There are 20 people in a room. A number of people have shaken hands. Prove that at least two people have shaken hands with the same number of people (Multiples handshakes between the same two people only counts as 1)