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Tuesday, January 31, 2006

The Sudoku/Systems connection

Epiphany:
A comprehension or perception of reality by means of a sudden intuitive realization (definition 3b in the American Heritage Dictionary)
So I guess I was visited by an epiphany this weekend: I realized that Sudoku has a surprising connection to systems of equations. Of course I’ve written separately about each of these — Sudoku on July 24 and August 6, systems on January 29 — but I hadn’t ever thought about any connection between them.

Now, however, as we’re starting our Algebra II unit on systems of equations, I have been thinking about how we characterize different kinds of systems. As is traditional, the Mathematics Teacher article that I described on January 29 characterizes any system of equations as having exactly one of the following descriptions:
  • consistent independent
  • consistent dependent
  • inconsistent
You may wonder why we don’t have four possibilities, splitting “inconsistent” into “inconsistent independent” and “consistent dependent.” Dr. Math gives a fairly good explanation:
“Inconsistent” means no solution. Independent and Dependent both mean there is a solution, so they can’t ever go with Inconsistent because that would be contradictory.

So really there are only three possibilities: Consistent Dependent, Consistent Independent, and Inconsistent.

We ordinarily don’t even use “consistent” with dependent or independent, since once you know what these latter two words mean, you already know they are consistent, so it is enough to say the system is “dependent” or “independent.”

We usually use the word “consistent” when we are more interested in indicating that the system does have a solution, rather than indicating how many solutions it has.
OK, so how does all this possibly relate to Sudoku? What occurred to me over the weekend — while working on a Sudoku puzzle — was that the process of filling in any given cell in Sudoku yields exactly the same three possibilities, and with the same meanings. Consider, for example, the following fragment of a partially completed puzzle (showing the first row and the first column):



From the first column alone, we can determine that the upper-left cell must hold a 5. But the first row gives us no additional information; thus this system is dependent. On the other hand, consider this slightly different variation:



Both the row and the column are needed in order to conclude that the first cell holds a 5 and the last cell in the first row holds an 8.

Finally, if you make a mistake in solving a Sudoku — or if the puzzle was constructed or printed incorrectly — you can end up with an inconsistent system:


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