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The Science Behind the Sudoku Puzzles

A game of logic especially as complicated as that of Sudoku puzzles is more often appealing to people who loves technical stuff like computer geeks, mathematicians, computer engineers, etc. It is quite fascinating how Sudoku captured the hearts of millions.

The other highly logical game that gained the same popularity is the Rubik’s cube which uses a three dimensional grid that requires you to group colors on each side. A Sudoku puzzle however is a flat grid that normally contains a nine row by nine column grid. It consists of 81 cells and nine smaller squares or sub-grids where numbers are placed.

A set of numbers from 1 to 9 are places in each squares with a digit appearing only once in every row and every column which is actually the literal translation of its Japanese name that means “unmarried number”.

Each puzzle has one unique solution. Although you may see numbers it does not require you to use any form of mathematical equations. This grid is known to be a special type of Latin squares which is credited to 18th Century mathematician Leonhard Euler where a formula of n x n matrices are filled with n symbols with each symbol appearing only once in each row or column. A Sudoku puzzle is c combination Latin squares with an added requirement of treating the puzzle as a single grid but with the sub-grids also containing the digits 1 to 9 that forms an interaction and restrictions with the other grids.

It seems like Sudoku is just a form of entertainment for most people but for mathematicians, it raises a whole lot of questions that required research. One of the questions raised about it is how many Sudoku grids can exist. To simplify this problem, mathematicians focused only the use of logic first was able to come to an estimated value of 6,670,903,752,021,072,936,960 possible valid Sudoku grids. This is based on a study by Bertram Felgenhauer of the Technical University of Dresden in Germany and Frazer Jarvis of the University of Sheffield in England. This value has been verified by other studies a couple of times already.

However if we only count those of the grids that can be reduced to equivalent configurations once, then we arrive at a smaller value of 5,472,730,538. These numbers assure mankind that we will not run out of puzzles to solve and we can continue to enjoy solving Sudoku puzzles for the rest of our lifetimes.

Another problem that is currently baffling scientists is what is smallest number of digits or symbols a puzzle maker can put into a starting grid in order to arrive at a puzzle that only produces one solution. Gordon Royle ofthe University of Western Australia was able to collect a total of 38,000 examples that fit into this criterion of having 17 as the answer to this question but still cannot be translated into one another if elementary operations are performed.

It should be noted that the question of what is the maximum number of digits can be placed in a starting grid and still get only one solution to the puzzle has not yet been arrived at yet. These are just some of the problems that still confuse and challenges scientists about Sudoku.

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