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Is Sudoku Mathematics?   (Is composing mazes mathematics?)

     An essay in the Monitor in October 2005, commenting on the new Sudoku puzzles, asserts they require no mathematics skills.  I concede that they require no arithmetic, but the logical reasoning that they require is certainly mathematics.  During a career as a college professor of mathematics, I enjoyed distinguishing between math and arithmetic, even when guest lecturing to, for example,  nine-year-olds.  

      Let me offer a simple example:  here is a method for constructing  mazes.  As a child, I loved mazes, but couldn’t make one myself without drawing the answer first and then drawing the maze around the answer - not very satisfactory, as the answer was then already filled in!  I didn’t realize the essential trick until graduate school.  Take a paper and two colored pencils (or a pencil and pen)  and follow along.

      Draw a rectangular  box (with the pen if you have one), with the upper left corner open for an entrance and the lower right open for an exit.  Now add lines to the maze as follows: you can put in any sort of a line you want - straight, curved, turning corners - provided that each line you add touches exactly one line that was there already.  (It cannot touch zero, it cannot touch two.)  
<<Incidentally, the maze need not be a rectangle - see the diagram below.>>

      When you have the box full enough with lines, stop.  I claim that the result is a maze - maybe an easy one, maybe a hard one.  It has exactly one solution - one path through from the start to the finish.

       Why am I sure there is a solution, and just one solution?  Let’s draw the maze again, using two colors this time   (we’ll call them blue and red, but any two colors - or a pen and pencil - will do.)   When you draw the box, make the top and right edges blue, and the bottom and left edges red   (so you have one blue line, with a corner, and one red line, with a corner.)  Now when you add new lines to the drawing, each new line touches exactly one old line - make it the same color as the line it touches.  The red and blue parts of the picture grow like spiders or tangles of spaghetti , getting tangled together, but there are always two parts, a red part and a blue part. And when you are done, there is one path through the maze - just start at the beginning and keep the blue on your left and the red on your right, and you’ll find the path through!

       This is clearly mathematics - it depends critically on the difference between “two” and “not two.”  But it is not arithmetic (and neither is Sudoku.)   

       I’ve had a class go out to the playground at recess, teach this to the class next door, and have that class go in and show it to their teacher and convince the second teacher that the method works.  But I didn’t let on that they were reproducing a “theorem” and a “proof,” which I suspect might have scared them off.  I like mathematics, and suspect some other people do to - maybe especially if they don’t know if they are doing mathematics.

Edward Ordman   (C)  2005

 Several mazes illustrating the text  

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