For each of the regions below: Can it be tiled with 1x2 dominoes, and if so, is the tiling unique?
For the solutions that follow, the number on the domino is the stage at which its position was forced. That is, the position of each domino labelled 1 was forced immediately. After all the dominoes labelled 1 were placed, the position of each domino labelled 2 was forced, and so on.
The first puzzle has two solutions. Notice the unmarked squares could be tiled by solely horizontal dominoes, or solely vertical ones. Thus, there are two possible solutions.
The second puzzle has one solution. The position of every tile is forced.
The third puzzle has no solutions. Notice the two squares labelled with sixes below. Each can be tiled by solely horizontal dominoes, or solely vertical ones. However you choose to fill those squares, the unmarked squares below cannot be tiled after, and so this puzzle must have no solutions.
This puzzle comes from "The Inquisitive Problem Solver."