Simple Proof of the Tetris Lamp
Jack Morris (via Hacker News):The lamp itself is composed of 7 individual pieces, containing a total of 28 squares. Therefore, assuming we can indeed form it into a rectangle, it would have to be 7x4 or 14x2 squares in size. I’m using the former case here simply because it’s a more natural shape, however this proof applies equally as well to the latter. Now imagine that we label each of these squares with a colour - either black or white - such that they form a checkerboard pattern as shown above. Notice that the number of black squares must be equal to the number of white, a property we’ll exploit. So that’s 14 black squares, and 14 white. Looking at each of the pieces individually, the issue with our assumption quickly appears.
