Brainteasers

Some interesting problems at various levels of impossibility:

• Think of a chessboard, can you cover the board in dominoes so that each domino straddles two squares exactly?
• Can you do it for other sized boards? Not just 8x8 but nxn?
• Can you do it if you take out the bottom corners and just try to cover the rest of the board?
• Can you do it if you take out diagonally opposite corners?
• Think of a chessboard which has a number of squares on each side which is a power of two ie 2x2x2x2x.... some amount of times. Take out any square at random. Can you cover the rest of the board in trionimos, L shaped blocks made of three squares?
• Two trains are on the same track, each is moving at a constant 30mph and they are a mile apart on a collision course. A super-fly starts out at one train and rushes towards the other one at 100mph, once there it turns round instantly and rushes back to the other at 100mph. How far does the fly travel before it gets squashed between the trains?