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The mathematics behind every puzzle

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The Mathematics Of Sudoku

To learn sudoku you need to study all the different techniques available or you will have no hope whatsoever in
becoming the best you can be at this wonderful game. Today we are going to take a quick look at the mathematics
behind it.

Unlike other puzzles sudoku is based on a structure and to solve it you need to have at least some level of logic.
The basis behind solving a Sudoku is what's reffered to as NP-complete, and it's called that because it is solved
on n2 x n2 grids of n x n cells.

It is because of this that people who are trying to learn sudoku find it difficult to solve. When you put cells on grids and
than throw in a few numbers it will take some logic and reasoning to correctly solve the puzzle.

Sudoku also has a game tree. The game tree is rather large and when only one solution is possible, it will make
solving the puzzle fast very difficult. There are fortunately, tips and certain little known strategies that you can use
to learn sudoku as fast as possible.

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A Latin Square

The Sudoku grid is very similar to a latin square. A Latin square is an n x n table that is filled with different symbols

so that each symbol occurs only once in each row and once in each column.

Here are a couple of examples.

Though note there are less solution grids available for Sudoku, than there are Latin squares.

The reason for this is that sudoku has multiple regions (3 x 3 in standard sudoku), where as with latin squares you're only working with one. To solve sudoku each 3 x 3 mini grid must also contain the numbers 1–9. Look at the example below.

Even so there are litterally endless solution grids for the Sudoku puzzle.

In 2005 a man by the name of Bertram Felgenhauer calculated the number to be about 6,670,903,752,021,072,936,960.

It has not yet been calculated how many solution grids there are for the 16 x 16 Sudoku puzzle. There are some 9 x9 grids

that can be recreated into other grids.

This can be done by

(1) rotating the grid,

(2) permuting some columns and rows

(3) changing around the numbers.

In 2005 Frazer Jarvis and Ed Russell calculated the amount of different Sudoku grids to total of 5,472,730,538.

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