# The Scanning Technique in Sudoku

#### Description

Mathematician Laura Taalman demonstrates the scanning 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 discussing how to solve Sudoku puzzles. In this clip, we'll talk about one of the most powerful and useful Sudoku solving techniques known as Scanning. With the One Place technique, we looked at rows or columns and try to find the one place where a given number could go. If we were lucky we narrow down to one place by looking at the restrictions caused where other rows, columns or blocks intersected the row or column we are interested in. Let's do the same thing, but starting with a block of the Sudoku puzzle instead of a row or a column. By scanning the rows and columns that intersect this block, we can sometimes narrow down to one place in our block for a given number. For example, consider the upper left block and look at the rows and columns that intersect this block. There must be a one somewhere in our block. Where? Well, the second row already contains a one, so we know that one cannot appear anywhere in our block that is intersected by that row. The presence of the one in the third row crosses out three more locations in our block. The third column blocks out one more; this leaves only one place for the one in this block. Scanning is an incredibly powerful solving technique. One way to start solving a Sudoku puzzle is to first scan for as many ones a possible, then for as many twos as possible and so on. With this in mind, let's keep scanning for ones. For example, the center block does not yet contain a one. We need to scan for ones in the three rows and three columns that intersect this block. This immediately crosses out all but one possible location for the one in this block, so we can place the one. Now, the lower right block is the only block in the Sudoku puzzle that does not contain a one. By scanning in the rows and columns that intersect this block, we immediately see that there is only one place that we can put a one. Now, we have placed all the ones; notice that they are distributed in the non-attacking pattern that we discussed earlier. This type of pattern is what you are scanning for when you use this technique. Let's try the same thing again, but this time scanning for threes. In this particular puzzle, we are actually be able to find all of the threes this way right now. This would be a good place to press pause and see if you could find them yourself. In the upper middle block, scanning reveals only one possible location for a three. Likewise, in the left middle block, in the center block and in the lower right block. We found all the threes; notice again there, non-attacking pattern. Now, that we have placed all the ones and all the threes, you might find it helpful to place small checkmarks above one and three in the top row to remind yourself that you don't have to worry about those numbers anymore. Now, we only have to worry about the other seven types of numbers. We never have to think about ones or threes again in this puzzle. Now, that we know how to use Scanning we are really making some progress; but, what can we do when just simple scanning isn't enough? In the next clip, we'll talk about an even more powerful technique called Double Scanning used in Sudoku.

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