Math’s ExplorationThe Math Behind SudokuRationale and OverviewSudoku is a logic-based combinatorial number-placement puzzle and there really no tricks or twists built into it. The objective of the game is to fill a 9×9 block with number so that each box in each column or row of the nine 3×3 sub grids that compose the grid contains all of the digits from 1 to 9. There are different kind of sudoku with different size of grid. The game current form was invented by American Howard Garns in 1979 and published by Dell Magazines as “Numbers in Place.”Unlike the crossword, however, sudoku is a game that required a great amount of focus more than any other puzzle game. Moreover, it doesn’t require any knowledge to solve it like the crossword puzzles. This game never ends unlike computer games which after several days of playing finishes it has infinite different stages. Also, there are many solutions to a single sudoku, so you don’t get bored of playing it again and again.Sudoku become my hobby as I saw it on the newspaper and just attempt to solve it. I began to enjoy its more as its give the satisfaction that comes from solving a puzzle seems to provide “relief” from stress in school. This may be just me but it would seem to explain my puzzle crazes. As I try to solve the puzzle, I began to wonder how many number of classic 9×9 Sudoku solution grids are these by using permutations because its seem that newspaper never run out of sudoku game. I also want to know if it’s possible for a traditional 9×9 sudoku to have multiple solution.Rule and basic strategy The rules of the game are simple: each of the nine blocks must contain all the numbers 1-9 within its squares. Each number can only appear once in a row, column or box. The difficulty is that each vertical nine-square column, or horizontal nine-square line across, must also contain the numbers 1-9, without repetition. The most basic strategy to solve a Sudoku is to fill in each cell all possible entries that not contradict the other entries in other cell. If you end up with one possible entry for a cell, it is an entry that you should fill in. Another way to solve is to pick a number and a row, column, or block. Check all the cells in the row, column, or block in which the number can be fill in. If the digit can only be placed in one cell in the block, you should fill that cell in. Once you’ve done this, the chosen digit can be eliminated from any other cell in that block or row.Example: Using the elimination technique, we see that 7 can only be in square d7.These are the two most basic strategy to solve sudoku but for more complicate sudoku, you would need more complicated analysis methods to make progress.Number of combinations of Sudoku1) In a block or row, there is always a completed solution of one-to-one mapping of the numbers 1 to 9.Original: 1 2 3 4 5 6 7 8 9Mapped:     8 9 5 6 4 2 1 3 42) There are 9! mappings, producing 362,880 solutions. 9×8×7×6×5×4×3×2×1= 362,880 solutions3) Within each solution in the Original Set, it is possible to permute the rows and columns of each minor (3×3) block. For example, rows 1,2 and 3 can be permuted six ways, as can columns 1,2 and 3. Similarly, the other two rows and the other two sets of columns can be Permuted in the same way. These permutations can be combined in 6 to the power 6 to get the number of ways6^6 permutations= 46656 ways   There may some solutions that could be permuted in one of those 6^6 ways to get the same solution as if permuted the numbers in one of those 9! Ways but I think that it’s a pretty rare where this could not happen.4) There 362,880 solutions x 46,656 ways = 16,930,529,280 unique Sudoku puzzles.VerificationAccording to Bertram and Ed,6670903752021072936960 in total is the exact number unique Sudoku puzzles. It’s a much bigger number than my answer but it doesn’t mean my answer is completely incorrect. It’s just that not every Sudoku puzzle can be permuted in one of 16,930,529,280 ways into a single solution. In fact, there are many different solutions that cannot be permuted to one another. It’s also impossible to calculated without the computer program because much of computing this number was done on the computer.There are only a few steps but require complex calculation1)list all the possible top three rows 2)count how many ways the top three rows can be extended to a full grid. The actual number of possible configurations for the top three rows is 948109639680! and every configuration of the top three rows gives the same answer in the second step as one of just 44. Its mean that each of the 948109639680 configurations in top three rows can be transformed by a sequence of these transformations into one of the list of 44.Here is the list of the configuration:To get the final total, multiply the green number by the red number in each row, and take their sum. Then multiply by 1881169920(this accounts for relabeling and some other transformations which always exist) to get the final answer, which is 670903752021072936960 in total.ConclusionMy answer is 16,930,529,280 unique Sudoku and the correct answer is 670903752021072936960 in total. Even though I didn’t find the correct number of permutations of sudoku, its wasn’t completely wrong. It’s just that not every Sudoku puzzle can be permuted in one of 16,930,529,280 ways into a single solution. It’s would also take a lot of work to find even one solution in each collection of 16,930,529,280 ways.


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