# The Double Scanning Technique in Sudoku

#### Description

Mathematician Laura Taalman demonstrates the double 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 talking about how to solve Sudoku puzzles. In this clip, we are going to talk about the powerful Sudoku solving technique called Double Scanning. Recall that the method of scanning can narrow down the possibilities in the block of a Sudoku puzzle by using the information in the rows and columns that intersect that block. Sometimes regular scanning doesn't provide enough information to place a given number in a given block, but in some of these cases you can still recover enough information about the block to find the hidden restriction that will enable you to scan a second time. For example, suppose we try to use scanning to place an eight in the upper right block, we can narrow down the placement of the eight to two cells but we don't know which of these two cells is the one correct answer. However, both of these possible cells are in the same column and one of them must contain an eight. This means that no other eight can appear anywhere else in that same column. This gives us another scanning line. Now, we just have to figure out how we can use it. One other block in the Sudoku puzzle might this be useful for. The right middle block now has restrictions for the number eight in all, but one of its cells; this means, we that we know where to place the eight in this block. Since we scanned twice to find this eight, we called this method double scanning. Remember that you should always be on the lookout for further consequences. Each number you place is a possible clue to the remaining numbers. For example, the eight we just placed can help us scan for another eight in the left middle block, and this new eight leads to a very simple one choice in that same block, since all other numbers in that block are filled in except for one, the five. Can you find another place where we might be able to use double scanning? Look for a block where scanning for a particular number does not allow you to place the number, but does allow you to place the number in a certain row, or a certain column within the block. Let's try the upper middle block with the number tow. Scanning shows us that the two in this block must be somewhere in the last column of the block. This means that no twos can appear anywhere else in that column. Where does this help us? Double Scanning now tells us that the lower middle block has only one place for a two. Once again let's look for some consequences of our new entry. We are almost done with the middle column of the Sudoku puzzle. There are only two open cells and each must contain either six or seven. There is already a six in the territory of the lower open cell, so that cell must be a seven and the upper open cell must be a six. We've finished the column; with the techniques of one choice, one place, scanning and double scanning under our belts, we can solve most Sudoku puzzle of moderate difficulty. In the next clip we will use these techniques to finish solving the Sudoku puzzle we have been working on.

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