# The One Choice Technique in Sudoku

#### Description

Mathematician Laura Taalman demonstrates the one choice technique in Sudoku.

#### Expert

Laura Taalman

Laura Taalman is an Associate Professor of Mathematics at James Madison University. She received her Ph.D in mathematics from Duke University, and her undergraduate degree from the University of Chicago. Her research includes singular algebraic geometry, knot theory, and the mathematics of puzzles. She is the author of Integrated Calculus, a textbook that combines calculus, pre-calculus, and algebra into one course, and a recipient of the Trevor Evans Award and the Alder Award from the Mathematical Association of America.  As part of Brainfreeze Puzzles, she is an author of the puzzle book Color Sudoku.  Laura lives in Harrisonburg, Virginia where she spends way too much time playing and making puzzles.

#### Transcript

Laura Taalman: Hi! I am Laura Taalman from Brainfreeze Puzzles. Today, we are talking about how to solve Sudoku puzzles. In this clip, we'll discuss the most basic Sudoku solving technique, called One Choice. Remember the one rule of Sudoku, the numbers one through nine must each appear exactly once in each row, column and block. Each cell on the board has a territory consisting of the row, column and block that it lies in. Sometimes we can find a cell on the board, whose territory already contains eight of these nine numbers, leaving us with only one choice of number to put in that cell. For example, consider this cell and its territory. One of the numbers one through nine must go in this cell. Which one? Well, there is already a one in this territory, so we can't put a one in our cell. Actually, there is more than one one in its territory. There is a one in the row for the cell and also a one in the column for the cell; but once we find just one of these ones we know that there can't be a one in our cell. There is already a two in the territory for our cell, so we can't put a two in our cell. There is also already a three. There isn't a four anywhere in the territory for this cell, so it is possible that we could put a four in this cell; but without checking the other numbers, we can't be certain that this is the only choice. Remember, there is only one possible solution to this puzzle, so we have to be sure that each number we write in the puzzle is the only possible choice. There is already a five in the territory for this cell, so we can't use five; and six seven eight and nine are also already in the territory. Now, we are sure that four is the one choice for this cell, so we can fill in the cell with a four. Of course, it's not always easy to find the cell in the puzzle whose territory already contains eight of the nine possible numbers. Let's look in another cell. A quick scan of its territory reveals that only the numbers two and eight are possible for this cell; but since we don't know which of these two numbers is the one correct choice we can't use the method of One Choice here. Can you find another cell on the board where we can use One Choice? Try to look for cells that have lots of different numbers in their territories. Here is one such cell; the territory for this cell contains all of the numbers one through nine except for the number five. Therefore five is the one choice that we can put in this cell. Remember that every time we fill in number in our puzzle, that number becomes a possible clue to finding the remaining numbers. Since we are close to finding all the numbers in the fourth row of the puzzle, let's try to finish the row. We know that the two open cells must each contain either eight or nine, since every other number is already in the row; but the middle block of the puzzle already contains a nine and this middle block is part of the territory of one of our open cells. This means that the second open cell has only one choice and that is the number eight and now very obvious application of One Choice tells us that the remaining cell in the row must be a nine. We finished the row. Now, we know how to use the one choice technique when solving a Sudoku puzzle, but this technique won't work everywhere in the Sudoku puzzle. We need some more techniques in our toolbox. In the next clip we'll talk about another basic Sudoku solving technique called One Place.

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