Solving sudoku requires absolutely no math, however there are still a number of mathematical ideas present in the creation of the game. The main two concepts are combinatorics and group theory. A great explanation of the combinatorics is given on a website for a university in the Netherlands, Kraket. It states, “The Sudoku is NP-complete. There are a few ways to see this. The first one refers to the result that to complete a partial Latin square is NP-complete. Another approach considers the Sudoku puzzle as a graph coloring problem. Each of the 81 cells is a vertex of the graph. Two vertices associated with two cells in the same row, or in the same column, or in the same block are connected by an edge. Thus, each vertex is linked with (adjacent to) 20 other vertices, which makes a total of 20 x 81 / 2 = 810 edges in the graph. The problem is to color the vertices with 9 colors such that no two adjacent vertices get the same color. Graph coloring of this type is known to be NP-complete.”

Now, that can be pretty hard to understand, so here is a video that has a slightly different way of explaining it:

Group theory is referring to each 3x3 box in a sudoku grid. The idea here is that each of these groups can be taken and rotated 90 degrees and give a completely new, valid sudoku puzzle. But how many sudoku puzzles are there? Through extensive research and calculations, it was found that there are 5,472,730,538 truly unique sudoku grids. However, by using group theory, rotations, and replacements (i.e. taking one puzzle and switching the twos to be sevens, the sevens to be nines, and so on), that number jumps to 6,670,903,752,021,072,936,960 (6 sextillion, 670 quintillion, 903 quadrillion, 752 trillion, 21 billion, 72 million, 936 thousand, 960). That is more sudoku puzzles than there are stars in the universe. According to Britannica, “… If each of the approximately 7.3 billion people on Earth solved one Sudoku puzzle every second, they wouldn’t get through all of them until about the year 30,992.”

Many sudoku playing sites offer different levels of difficulty for someone to choose from. Easy would have the most numbers included, making it fairly easy to solve. A medium difficulty would have slightly less numbers, making it slightly more challenging. Hard would have the smallest group of numbers possible but still be able to be solved. But what is the fewest numbers that a sudoku grid can have while still remaining solvable? Through much research, the answer is 17. Researchers have been trying to find sudoku puzzles that only offer 16 initial values that can still be solved, but to date, they have not been successful.