The Unbearable Uniqueness of Sudoku

The next simplest rule after the rule of elimination (see the last entry) is uniqueness (another name would be the pigeonhole principle ). This principle says that if there's only one place (the pigeonhole) to put a number (the pigeon), it must go there.

Consider the following puzzle:

The rule of elimination can get us this far:

Here's where the rule of uniqueness comes in. Looking at the pencil marks in the top row, we see that there is only one place for a 6 to go. Since every number must appear somewhere in every row, we know the 6 must go there. Similarly, there is only one place for a 9 to go in the 2nd row, and one place for a 3 in the 5th row.

Filling in these cells, we can perform an additional round of elimination. Some more unique cells pop up, such as the 5 in the 5th row. Earlier, there were two possible places for a 5 in that row, but one of those places was the only place for a 3. Thus, the 5 must belong in the other cell.

Now uniquess gives us a 4 and a 9.  A bit more elimination, and we‘re done!