Mutilated chessboard problem
Imagine a large chess board with a bunch of squares on it, just like the board you play on. But in the mutilated chessboard problem, someone has cut off some of the squares from the chess board.
Now, let's say you have a lot of tiny rectangular tiles that you want to use to cover the chess board. But here's the tricky part: you can only use these tiny tiles to cover the chess board in such a way that each tile covers exactly two squares on the chess board, and the tiles must always cover two squares that are next to each other (either horizontally or vertically).
So if someone cuts off a square from the chess board, there's no way you can use the tiny tiles to cover that missing square. But the question is: can you still cover the rest of the chess board with these tiles, or is it impossible now that some squares are missing?
Turns out, there's a clever way to figure it out. You can color in all the remaining squares on the chess board with two different colors (let's say black and white). Then, if the number of black squares remaining on the board is equal to the number of white squares remaining on the board, it's possible to cover the board with the tiny tiles.
Why does this work? Well, each tiny tile will cover either one black square and one white square, or two black squares or two white squares. So if you have an equal number of black and white squares, you can always find enough tiles to cover each square with another tile.
It's a cool little puzzle, and a good exercise in thinking logically and spatially!
Now, let's say you have a lot of tiny rectangular tiles that you want to use to cover the chess board. But here's the tricky part: you can only use these tiny tiles to cover the chess board in such a way that each tile covers exactly two squares on the chess board, and the tiles must always cover two squares that are next to each other (either horizontally or vertically).
So if someone cuts off a square from the chess board, there's no way you can use the tiny tiles to cover that missing square. But the question is: can you still cover the rest of the chess board with these tiles, or is it impossible now that some squares are missing?
Turns out, there's a clever way to figure it out. You can color in all the remaining squares on the chess board with two different colors (let's say black and white). Then, if the number of black squares remaining on the board is equal to the number of white squares remaining on the board, it's possible to cover the board with the tiny tiles.
Why does this work? Well, each tiny tile will cover either one black square and one white square, or two black squares or two white squares. So if you have an equal number of black and white squares, you can always find enough tiles to cover each square with another tile.
It's a cool little puzzle, and a good exercise in thinking logically and spatially!