Center of a circle

Using only a compass, can you find the center of a circle?
NOTE: You can't use anything other than compass. Not even a ruler or a scale. This implies, you can't make a straight also.

Dominoes on a square board

Suppose you have cut off the two diagonally opposite corner squares of a regular 8×8 square board, as shown above, 62 squares remain. Now you have a set of 2×1 dominoes, each of which covers two squares that are adjacent vertically or horizontally. Is it possible to use 31 of these dominoes to tile all 62 squares?

13 coins

You are given 13 coins out of which one is defected. You don't know the defect 'heavy' or 'light'. Also you are given an authentic coin. You are allowed to use weighing balance 3 times. Give a strategy to find the defected coin and its defect.

NOTE: This is in continuation to the puzzle "Balls and Weighing Balance". If you are asked to only the defected coin, you can do so without the authentic coin also. But to find the defect, you need to figure out a new strategy making use of authentic coin.
Just, to avoid confusion, total coins would be 14(13 + 1authentic coin)

Grid Infection

In an n by n grid of squares, two squares are neighbors if they share an edge. Initially, some squares are "infected". At successive clock ticks, an uninfected square gets infected if at least two of its neighbors are infected. How many squares must initially be infected so that all squares eventually get infected?

Fox in a Hole

Consider 'n' holes in a line. One of them is occupied by a fox. Each night, the fox moves to a neighboring hole, either to the left or to the right. Each morning, you get to inspect a hole of your choice. What strategy would ensure that the fox is eventually caught?

Poison in Drink????

There was a booz party going on with many guests. Some of the guests left early after enjoying their drinks. While the others drank overnight. Next morning, news came out that everyone in the booz party died due to poison. Only those who left did not died. Figure out the source of poison when the same drink drank by the early leavers were able to survive. It has nothing to do with the quantity of poison consumed.