I Knew It: There Can Be More than One Solution to a Sudoku Puzzle!

 
A recent article by Agnes M. Herzberg and M. Ram Murty in the Notices of the American Mathematical Society confirms what many of us have known for years: there's more than one way to solve a Sudoku puzzle. Unfortunately, most of them are still wrong [Sudoku Squares and Chromatic Polynomials].

However—and here's the encouraging part—there are sometimes more than one right answer. The puzzle shown on the right is an example. Try it. (Answer below.)

There are two different ways to fill in the blank cells. Both of them are valid solutions.

I've always thought that the Sudoku puzzles were more like logic puzzles than mathematical puzzles and the article confirms that impression. The puzzles can be solved using colors instead of numbers. Here's what the authors say about the popularity of these puzzles.

It is interesting to note that the Sudoku puzzle is extremely popular for a variety of reasons. First,it is sufficiently difficult to pose a serious mental challenge for anyone attempting to do the puzzle. Secondly, simply by scanning rows and columns, it is easy to enter the “missing colors”, and this gives the solver some encouragement to persist. The novice is usually stumped after some time. However, the puzzle can be systematically solved by keeping track of the unused colors in each row, in each column, and in each sub-grid. A simple process of elimination often leads one to complete the puzzle. Some of the puzzles classified under the “fiendish” category involve a slightly more refined version of this elimination process, but the general strategy is the same. One could argue that the Sudoku puzzle develops logical skills necessary for mathematical thought.

How many of you are hooked?