Introduction to "Sudoku Strategies and the Sudokerson Matrix" The table of contents will give you some idea of what is contained in my book:
- The Sudokerson Matrix
- Level: Easy
- Level: Medium
- Level: Hard
- Level: Diabolical
- Beyond Diabolical
- Over the Top
- Theseus Lives!
- Sudoku Strategies
- A Field Test of Parallel Analysis
Appendix 1: Notes on Sudoku Design
Appendix 2: How Many Different Sudoku Puzzles Are There?The book outlines a four-step method for solving Sudokus: three increasingly valuable basic strategies (Chapters 2-4), followed by an even more powerful fourth tactic that I call "parallel analysis" (Chapter 5). The latter is the most effective Sudoku technique that I know of. It does more than simply lead you to the correct answer. One of the hypnotic aspects of parallel analysis is that it can be used to show you the effects of changing a particular digit, and even how to design your own Sudoku puzzle. I have found it especially interesting to choose a published puzzle that happens to have a starting digit in the central cell, and then to vary this digit through its other possible choices and watch what this does to the solution process and to the answers obtained (Chapter 6). Some of these can be pretty astonishing. Ferreting out all the different crashes and wrong answers, comparing them with one another and with the right answer, can be much more interesting than merely going for a single correct solution. Parallel analysis will take you straight to your goal if you like, but it can also suggest interesting side trips along the way.
Many years ago I heard the story of a small boy whose parents presented him with a toy drum for his birthday. He played the drum incessantly, and nearly drove his grandfather mad until the old man presented his grandson with a beautiful pearl-handled pocket knife, saying to him, "Here's a belated birthday present that you might like. And oh, by the way, do you know what is inside your drum?" Chapters 6 and 7 are devoted to what's inside your Sudoku puzzle, and parallel analysis is the knife.
Features that show up frequently when you commence dissecting and rebuilding Sudoku puzzles are ambiguous loops. Ambiguous loops are closed loops of cells with pairs of allowable digits arranged so they can yield more than one acceptable solution. Professional Sudoku designers abhor them, but they can be fascinating. I have been making a systematic study of how many, and how large, ambiguous loops there can be in a solvable Sudoku. The trick is first to map out the loop that you want to find, then surround it with digits that satisfy the Sudoku rules, and finally choose a starting set containing enough of those digits to permit the puzzle to be solved in the normal way. I call this backward analysis process "Kudosu," and am writing up the results in a previously-mentioned booklet entitled "The Sudoku from Hell...." which I would be happy to send upon request. In it are giant ambiguous closed loops made up of as many as 18 cells, each containing the same two digits. I have also determined that the maximum number of ambiguous loops possible for a given Sudoku matrix is four, and that a 16-cell ambiguous loop is impossible.
Serious Sudoku designers abhor such multiple solutions, of course, and discard any trial puzzle that exhibits them. But to me they are fascinating; they take one onto new ground and into strange territory. Some of the solutions can be quite beautiful examples of abstract art. Of course you do not necessarily have to become involved with ambiguous loops and multiple solutions. My four-step strategies are a powerful way of solving normal Sudokus. But the point is that the strategies are so powerful that you can alter, explore and design puzzles on your own, in a way that would be difficult and tedious by conventional methods.
Finally, in an appendix to "Sudoku Strategies...." I consider Brian Hayes' recent study of just how many different valid Sudoku matrices actually exist. The total number of different arrangements of nine digits in a 9x9 matrix of 81 cells is 9 to the 81st power, or roughly 200 thousand trillion trillion trillion trillion trillion trillion. But after imposing all the boundary conditions of Sudoku, including interchanging names of digits and rotating and reflecting the matrix, the number of genuinely different Sudoku answer matrices possible falls to a mere 5.5 billion. Suppose that you could solve four Sudokus per hour, 24/7/52, without stopping for food, sleep, or bathroom breaks. How many years would it take you to solve them all? The answer is in my book "Sudoku Strategies...."
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