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 7×28 rectangle so that they are all used and all inside the rectangle? The closest I have managed is this:
Source: Puzzling StackExchange
No, it is impossible. Imagine you are placing the tetrominoes on a 7×28 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 an equal number of black and white cells.