Adam Gauthier
Prof. Winkler
Math 7
April 11, 2013
My Favorite Puzzle
Growing up, I wasn’t always interested in figuring out puzzles. I got
stumped by them very easily and would give up too soon. One puzzle stood
out to me: it was challenging, but I could also be successful. That puzzle was
Sudoku.
Sudoku isn’t a very challenging puzzle to learn how to do, but its
simplicity is deceiving. Sudoku is a simple, logic-based number placement
puzzle. The rules for the puzzle are actually quite simple; the puzzle grid is
given to you and contains 81 cells (the whole thing is 9x9), which is divided
into nine columns, rows, and regions. Each 3x3 region is also bordered
within the larger grid. The task is to place the integers from 1 to 9 into the
empty cells in such a way that in every row (9 cells long), column (9 cells
long), and region (3×3 cells) displays each number only once. The puzzle
begins with a few of the numbers already filled in. Most normal level
Sudoku puzzles have anywhere from 22 to 30 cells filled out before you start
to give you a few starter hints. The integers given are placed very
strategically so as to provide help, but also make it more difficult to
complete the puzzle down the road if you make an error. Very often in
Sudoku, you don’t find out the mistake you made until after you start filling
in many more of the cells; at that point it is often best to start over
completely. The key to Sudoku is logic; there isn’t much mathematical skill
needed to solve it. There isn’t any addition or subtraction. No multiplying or
dividing. So while it is a mathematics puzzle, it is more a test of your logic
than your mathematical prowess.
The puzzle has a long story behind it. The origin of the Sudoku puzzle
can be traced back to Switzerland. Leonhard Euler, a Swiss mathematician
and physicist, created a puzzle called "carré latin" back in the 1700s, which
is similar to today’s Sudoku puzzle, but without the additional restriction on
the contents of specific regions. Howard Garns, an American architect and
the creator of today’s Sudoku puzzle, didn’t publish the first real Sudoku
puzzle until 1979. The puzzle really didn’t pick up popularity until about
1986, after it was published again and given the name Sudoku by Nikoli, a
Japanese publisher that specializes in games and, especially, logic puzzles.
Nikoli became prominent worldwide with the popularity of Sudoku. Sudoku
was also far more popular when it was republished in Japan. The name
given, “Sudoku,” is short for the term Su-ji wa dokushin ni kagiru, which
means "the numbers must be single”.
For such a simple concept, the number of possibilities is vast. There
are over 6.67x1019 possible Sudoku combinations in the 9x9 board. When
you add to that the different starter hint options that can be provided, the
potential for repeating the same puzzle twice is highly unlikely, if it is
randomly generated.
Sudoku puzzles can be solved in several different ways, and there are
many different philosophies about technique. It is usually a combination of
techniques that brings success. The basic logic is to narrow down the options
so that a candidate for a box can be ruled out in all the other
row/column/grid options related to that box. Additionally, that one piece of
information gives you clues about the other rows and columns that are
related to that grid. A search on the Internet shows a countless number of
theories, mathematics, logic, and tricks. There are names, analysis methods
(X-Wings, Swordfish), rules (naked tuples and hidden tuples), and plain old
uniqueness.
I use a method called cross-hatching. I begin any Sudoku puzzle by
identifying which number occurs most frequently. I look for regions without
this number and try to place the number in a fitting spot by process of
elimination. For this purpose, I identify which columns and rows contain the
first number chosen, because the number cannot be placed in any of these
rows/columns/regions. If there is only one feasible place left in a region, the
number can be entered in the spot. I repeat the strategy with a different
number, usually the number that shows up the second most frequently. Once
this has been repeated a few times, it becomes more about finding which
numbers complete the 1-9 sequence, given the rules. First, I check every
region for the needed number, then determine into which cells the number
doesn’t fit. From there, I “mark” which spots are available for that number in
the region, and move on to the next region, or the next number. As parts of
the puzzle are completed, the rest goes quickly. It becomes all about what
numbers can and can’t fit, and verifying that any new number you enter is a
match for each row/column/region that it touches.
I enjoy working a math or logic puzzle when my mind gets
overwhelmed during studying, or when I need to get ready to study. The
logic and method used to solve the puzzle is a balancer to chaos and it puts
me in a state of mind where I work more efficiently.