Math IA Intro When you play the game Sudoku, do you think about the math involved with it? The everyday person usually does not. The grids used in the game of Sudoku are made through the manipulation of numbers so that in the end, there is only one answer.
Without using math to make the grids, this popular game would not exist, at least not in the form it is today. For one, the creator of the game would not have been able to do so. Also, there would be no way to solve this puzzle so that there is only one answer. Therefore, math is an essential part of the building blocks in this popular puzzle. In this exploration, I will find the math behind the making and solving of the Sudoku puzzle grids. Researching theories and ideas and then actually testing them myself is the best option for me to find the solutions needed to complete this investigation effectively.
In my research, I have found that permutations, arithmetic analysis, and Gödel numbers are essential for the making and solving of the grids. Using these methods are necessary to produce the right numbers in the right places on the grid. I chose to investigate this question because I have always been interested in finding the math behind everyday items. This subject is my passion and being able to incorporate it into my life is something that intrigues me. Puzzles that incorporate math are, in my opinion, interesting and can be both simple and complex. I wanted to apply this to an extremely common everyday item, such as the game of Sudoku, to further my knowledge of how math influences people on a daily basis.
Background What is Sudoku? Sudoku is a logic-based placement puzzle, also known as Number Place in the UnitedStates. Although first published in a U.S.
puzzle magazine in 1979, Sudoku initially caught on in Japan in 1986 and attained international popularity in 2005. Numerous variations of Sudoku have been developed, with different shapes and sizes: 5×5 grids, 8×8 grids, 16×16 grids, 25×25 grids, grids with multiple overlapping blocks, and even a three-dimensional cube.Sudoku actually means “the numbers must be single,” or “the numbers must occur only once.” In fact, this famous puzzle’s name is actually short for “S?ji wa dokushin ni kagiru,” and the Japanese publisher, Nikoli Colonel Lieutenant, made the name Sudoku his trademark. Although this game is typically seen consisting of numbers, it can also contain letters or symbols. While there are mathematical ways to solve the Sudoku puzzles, one does not need to have any type of prior knowledge of math to successfully complete them. Also, there are benefits from playing these puzzles, other than the satisfaction of completion, such as “prevention of memory decline; and development of reasoning, analytical and logical thinking skills.” This makes Sudoku good for the mind, and for some people, good for entertainment; which makes this puzzle popular.
Investigation The game of Sudoku is merely different versions of permutations in its own way. A permutation is an “often major or fundamental change (as in character or condition) based primarily on rearrangement of existent elements.” There are nine rows, nine columns, and nine groups of nine numbers that make up this grid, each of which only contain one of each number that is one through nine.
An example of a blank Sudoku grid is shown below. To find out how many permutations there are in a standard set size of nine, you use the factorial of nine (9!). This number is 9x8x7x6x5x4x3x2x1, which is equivalent to 362,880 different ways to order the nine numbers in a row, column, or grouping. A property of permutations is arithmetic analysis, which is in the puzzle of Magic squares.
This puzzle and Sudoku are closely related. In the game of Magic squares, adding up the numbers will always give the same result. The same is for a standard set of Sudoku. If one adds or multiplies up all of the numbers in a row, column, or group, the answer is always 45, or 362,880. This becomes helpful when there is one number missing because then one can add up all of the numbers and subtract it from 45, or use the multiplication route, and that will give the missing number.
However, this does not necessarily work if there are more than one number missing from a row, column, or group.Another helpful strategy to the game Sudoku are Gödel numbers. This helps to eliminate the problems with missing numbers.
With Gödel numbers, instead of multiplying the Sudoku numbers, the prime numbers that correspond with the Sudoku numbers are multiplied. The corresponding prime numbers with the Sudoku numbers are in the chart below. An example of how the Gödel numbers work are if there is an incomplete row with the numbers 9, 6, 2, 8, 1, 3, 5, and 7, the conversion and then multiplication of their corresponding prime numbers is 23x13x3x19x2x5x11x17 = 31,870,410. The total is supposed to be 223,092,870, so by dividing that number by the 31,870,410, the number is seven, or in Sudoku numbers is 4.
“Sudoku grids are simply special cases of Latin squares… and a Latin square of order n is an n × n square containing each of the digits 1,…, n in every row and column.” The creators had to find out which gird answers actually worked and “the number of 9 × 9 Latin squares is 5524751496156892842531225600 ? 5.525 × 1027, therefore there was a large amount of time required to construct the right answers for the grids. Latin squares do not have a known general formula, making the calculation of not only them to be difficult, but also the calculations of the Sudoku grids as well. In the creating of the Sudoku grids, there was an extensive amount of math and time that led to the game that is widely known today.
Along with the standard 9×9 Sudoku Puzzle, there are different puzzle sizes as well. There is a 4×4 puzzle, a 16×16 puzzle, rectangular puzzles, and other larger and smaller sized puzzles. In the 4×4 puzzle, there are 16 total squares, making this puzzle generally simple to solve. “The minimum number of squares that can be revealed and still produce a solvable puzzle is four.” Similarly to the 9×9 puzzle, to solve this puzzle one uses the factorial of four (4!). This number is 1x2x3x4, which means that there are 24 ways to order the four numbers in a row, column, or grouping. A blank example of a 4×4 Sudoku puzzle grid is shown below.
Another, but more difficult Sudoku puzzle grid, is the 16×16 puzzle. This grid consists of 256 squares, and also has symbols along with numbers. Even though to some people this puzzle may seem extremely difficult, the different strategies that one uses for solving the 9×9 puzzle can also be used to complete the 16×16 puzzle. An example of a 16×16 puzzle is on the right below. There are also numerous Sudoku puzzle grids called Rectangular puzzles.
“A set of same sized rectangles can be arranged into a square puzzle grid.” There are countless ways to arrange such a grid, and “the only grid sizes that can’t be used are those that are prime numbers and so can’t be divided up into a rectangular block (e.g. a 5×5 puzzle can not be subdivided in any different way).
” An example of a Rectangular Puzzle is on the left below.ConclusionWhile there are innumerable amounts of Sudoku puzzle grids, people can use the same types of strategies to solve them. Since the first Sudoku was published in 1979, people have found that using the following strategies help to solve the Sudoku puzzles: permutations, arithmetic analysis, and Gödel numbers. These tactics are useful for effectively and mathematically completing Sudoku puzzles with the least amount of errors. Through using those strategies, someone who enjoys Sudoku puzzles will not only be entertained, but they will also get the benefits of intellectually engaging their brains. Bibliography”Blank Sudoku Grid for Download and Printing.
” Puzzle Stream,puzzlestream.com/sudoku/blank-grid.php.Buga, Natalia. Let’s Do Sudoku: 6 Illustrated Solving Techniques Plus 100 Hand-Crafted PuzzlesSpiced Up with Wise Quotations.
IUniverse, 2009, books.google.com/books?id=yM3M3l-hIJwC&pg=PA1&lpg=PA1&dq=math behind the 16×16 sudoku grid&source=bl&ots=phnTgnZ4qY&sig=dWGWQNySvfNp4gx80TRi5xC7pBM&hl=en&sa=X&ved=0ahUKEwjU7uLogpXYAhVP0WMKHewrAY8Q6AEIVTAM#v=onepage&q=math%20behind%20the%2016×16%20sudoku%20grid&f=false.Felgenhauer, Bertram, and Frazer Jarvis. Mathematics of Sudoku I. University of Sheffield, 25Jan. 2006, www.
pdf. “Sudoku Puzzle Theory.” Sudoku Puzzle Theory, www.sudokudragon.com/sudokutheory.htm.”The Math Behind Sudoku.” The Math Behind Sudoku: Counting Solutions,www.math.cornell.edu/~mec/Summer2009/Mahmood/Count.html.