1. One hundred hats
A prison guard tells his 100 prisoners that they will be playing the following game. The guard will line up the prisoners single-file, all facing the front of the line. The guard will place a red or a black hat on each prisoner’s head. There could be any combination of red and black hats (perhaps 50 and 50, perhaps 99 and 1, perhaps 100 and 0, etc.) and the hats could be in any order. Each prisoner can only see the hats of those in front of him in line, and not his own hat or those hats behind him. The guard will start with the prisoner at the end of the line (who can see all hats except his own), and ask “Is your hat red or black?”. If the prisoner responds correctly, he will be set free. Otherwise he will remain in jail. The guard will then ask the second to last prisoner in line the same question, with the same consequences. The guard will keep moving forward, asking the same question, until all 100 prisoners have been asked. Before the guard begins the game, the prisoners are allowed to devise a strategy. Once the game begins, the prisoners cannot communicate, except by answering “red” or “black” in response to the guard. They can’t use volume, intonation, or pauses in their response to communicate, as the guard will notice all these tricks (it’s a math riddle). Your task is to devise a strategy so that at most one prisoner remains in jail, and so that at least 99 prisoners are set free. How do you do it?
2. Three hats
There are 3 prisoners in a room. Two are wearing red hats, and one is wearing black hat. Each prisoner can see all hat colors except his own. The prison guard tells them, “Each of you is wearing a red or a black hat. Tell me your hat color, and you will be set free.” Since none of the prisoners can see their own hat, nobody responds. Next the guard adds, “At least one person is wearing a red hat.” Soon afterwards, a prisoner correctly identifies his hat color. How?
3. Tiling a chess board
(Part I) Cover the top right and bottom left squares of an 8 × 8 chessboard. Can you tile the remaining 62 squares with 31 tiles of size 2 × 1?
(Part II) Cover any 3 corner squares of a 30 × 30 board. Can you tile the remaining 897 squares with 299 tiles of size 3 × 1?
4. One hundred quarters
There are 100 quarters lying flat on a table. Twenty have heads facing up and the rest show tails. You are blindfolded and gloved so that you cannot see or feel which are are heads and which are tails. You are allowed to pick up and move the quarters however you like, but at the end of your manipulations each of the 100 quarters must be showing either heads or tails. Your task is to split the 100 quarters into two groups that are guaranteed to have the same number of heads facing up. How do you do it?

I'm so sorry if you cant find the answers..

-NIK-