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* An 8x8 chessboard can be entirely covered by 32 dominoes of size 2x1. Suppose we cut off two opposite corners of chess (i.e. two white blocks or two black blocks). Prove that now it is impossible to cover the remaining chessboard with 31 dominoes.
-* You are in a glass box looking at seven other copies of you in their own boxes. Each of you has a white or black hat, selected at random uniformly and independently, and you are all numbered from 1 to 8 in an arbitrary order. You can see everyone else's hat colors and numbers, but not your own. Each of you may select black or white. A total of $8000 is split between you all based on how many shares you have (if this would result in non-whole cents each share's payout is rounded down). Each of you starts with one share. If your color selection matches the color of your hat, you gain one share, but remove $400 from the total payout. If it does not, you retain one share and the total payout does not change. You may not communicate with each other. What do you do to maximize your return? What if you are given an option to abstain at a cost of $250 to you?
+* You are in a glass box looking at seven other copies of you in their own boxes. Each of you has a white or black hat, selected at random uniformly and independently, and you are all numbered from 1 to 8 in an arbitrary order. You can see everyone else's hat colors and numbers, but not your own. Each of you may select black or white. A total of $8000 is split between you all based on how many shares you have (if this would result in non-whole cents each share's payout is rounded down). Each of you starts with one share. If your color selection matches the color of your hat, you gain one share, but remove $400 from the total payout. If it does not, you retain one share and the total payout does not change. You may not communicate with each other. What do you do to maximize your return? What if you can abstain at a cost of $100 to you?What if you lose half the total payout if everyone abstains?
* There are 51 ants sitting on top of a square table with side length of 1. If you have a square card with side 1/5, can you put your card at a position on the table to guarantee that the card encompasses at least 3 ants?