Woxidu

A good friend from way back sent this one to me. It’s got two pretty fun solutions, one that I came up with and one that he showed me later on. Enjoy!

You’re a carpenter in a rectangular room and you have a bag of tiles, some are 2×2 and some are 1×4. Using all of the tiles in your bag the floor is completely covered (no gaps on the floor, no tiles left over). Just when you finish tiling, you drop your hammer and break one of the tiles. As you look around your supply closet, you find just one more tile, but it’s the wrong kind (if you broke a 2×2, then you found a 1×4 and vice-versa). Is it still possible to completely tile the room with this replacement? If so, under what circumstances? If not, why not?