SUDOKU STRATEGY HOW TO

To estimate how many ways of filling in the board, there can at most be, we multiply the maximal number of ways that there are of filling in each of the cells. Each entry of the table is no smaller than the number of ways of filling in the corresponding cell on a Sudoku board if you fill in the panel from left to right, top to bottom. To understand the idea for completing activity one better, we could draw up a 9×9 table like the following one. Use the fact that each row can contain each number at most once. Improved CountĪctivity 1: Modify the argument from above to show no more than (9x8x7圆x5x4x3x2x1)^9 = 362880^9, i.e., about 1.1×10^50 different 9×9 complete Sudoku grids. Mathematicians call an estimate like ours an “upper bound”: we have not computed the actual number of complete 9×9 Sudoku grids, but we have shown that there are no more than 9^81. Let’s put our result into context: we have only shown no more than complete 9^81 Sudoku grids of size 9×9. That’s not too far from the number of atoms in the universe (Observable Universe). A useful calculator will tell you that 981 is approximately 2×10^77, or, in words, two hundred quattuorvigintillion. Thus no more than 981 different complete Sudoku grids of size 9×9. We can continue the argument for all other cells, observing that there are at most 9^81 ways of filling in the first n cells. Therefore there are at most 9x9x9 ways of filling in the first three cells. For each of the 9×9 combinations we could choose for the first two cells, there are nine ways of filling in the third cell. For the third cell, we again have no more than nine various ways of filling it in. In total, that gives us at most 9×9 different ways of filling in the first two cells. For each of the possibilities that we could choose when filling in the first cell, there were no more than nine possibilities for filling in the second cell. Filling in the second choice, we have no more than nine options. Filling in just the first cell, we have nine choices. Now imagine filling the grid in, starting with the top-left cell and then proceeding row by row, from left to right. But let us ignore this detail and state that for each cell, there are no more than nine ways of filling it in. There will be significantly fewer possibilities in most cases since it is not allowed to have two copies of the same number in any row, column, or 3×3 box with a thick rim. Suppose we have an empty grid and fill it in, ensuring that each of the numbers from 1 to 9 appears exactly once in each row, column, and 3×3 box with thick margins.Įach cell must contain a number between 1 and 9, i.e., there are nine different possibilities. How many (complete!) 9×9 grids are there, i.e., how many different ways can one fill an empty 9×9 Sudoku grid while following the Sudoku rules? Have fun! Counting Sudokus A first look at counting Sudokus And, unless you have taken university-level Math or Computer Science classes before, mastery of some mathematical concepts for which you will find applications again and again in your life. What do you get out of all of this? A new way of looking at Sudoku puzzles, the right to claim that there is no Sudoku puzzle you can’t beat. You can see the outlines of solutions in section 6 and some further information sources in section 7.

I encourage you to attempt the activities that you can find throughout the sections. Section 5 explains how Sudoku puzzles are related to graphs and why this connection is so useful. At the same time, the other is more complicated but enables a human being to solve any Sudoku puzzle with a solution. One of them will be relatively straightforward in theory but impossible to use without a computer. In sections 3 and 4, we look at solving algorithms for Sudoku puzzles. Section 2 is the beginning of a discussion on the existence and uniqueness of answers to Sudoku puzzles. Just like a Sudoku solver presented with an incredibly hard puzzle, a Mathematician will be interested in when puzzles have unique solutions. What about N2 x N2 grids for any positive number N? And when will the New York Times run out of Sudoku puzzles? We will count Sudoku puzzles in section 1. You might wonder how many possible different 9×9 grids there are. By a Sudoku puzzle, we will mean a 9×9 grid puzzle like the one above, unless something different is stated explicitly.

SUDOKU STRATEGY HOW TO

On this webpage, we will not focus on how to solve the New York Times puzzle faster than the person sitting next to you (although we will learn some advanced solving tricks), but on aspects of Sudoku that are interesting from a mathematical perspective. Since you are reading an article on Sudoku puzzles’ mathematics, this is probably an easy exercise for you already.

Sudoku strategy