As a birthday present last year, I received some fridge magnets. They didn't come as a puzzle, so I don't know if they have a solution, but I made a puzzle out of them anyway. The magnets are tetrominoes. There are 7 of each shape. Is it possible to arrange them into a 7x28 rectangle so that they are all used and all inside the rectangle? The closest I have managed is this:
No, it is impossible. Imagine you are placing the tetrominoes on a giant chess board. All of them, except for the T-shaped ones cover exactly 2 black and 2 white cells. Each of the T-shaped tetrominos covers either 2 more black cells than white cells, or 2 more white cells than black cells. Since there are 7 of them, combined they will cover either more black cells than white cells, or more white cells than black cells. Therefore all pieces on the picture can not cover perfectly a rectangle, which contains equal number of black and white cells. However, every rectangle with even number of cells in it, e.g. 196, has as many black cells as white cells, which proves the impossibility.