Process of Elimination

When I first saw a Sudoko puzzle not long ago, my programming instincts kicked in right away. While some people may use intuition to solve puzzles, my tendency is to try to work out an algorithmic approach — something that‘s guaranteed to work every time, even if only by plodding along.

So, how can a puzzle be solved algorithmically? The key is to make progress towards the solution, one piece of information at a time. What information do we have about a puzzle? Of course we know the values of some of the cells. But even in a cell whose value we don‘t know, we can say that certain values are impossible because they would duplicate the value of another cell in the same row, column, or box. Looking at it the other way, if we know the value of a cell we can use elimination to remove that value as a possibility in all of the cells in the same row, grid, and box.

We can represent the state of our knowledge using pencil marks — little notes in each cell that indicate what values are possible for the cell. Of course, some people may prefer to mark values that have been rejected, or to make other sorts of marks, but I'll stick to marks that show possible values.

Let's solve a relatively easy Sudoku puzzle — one that only requires the rule of elimination. First, the clues:

After performing elimination on all the known values, we have quite a few pencil marks:

For all that work, only two cells now have definite values that didn't previously (namely, the 5 in the upper-left box and the 8 in the center box). Performing elimination using these values gives us the 7 in the upper middle box:

The 7, in turn, allows us to uncover the 3 and 6 in the same box:

Those values allow us to complete two of the boxes:

Continuing, we are to eliminate the 4 from cell (1, 2), leaving only a 1 there. We can remove all the 6's from the upper right box, allowing us to fill it in completely. The 6 in cell (4, 4) allows us to eliminate the 6 in cell (8, 4), which in turn allows us to fill in a 2 there.  Removing 2 and 6 from cell (7, 4), we get a 9.  Now we can cross out 2 and 9 from cell (1, 4), leaving a 3.  Making sure we cross out all the numbers that can be eliminated from the newly solved cells, we get:

The 1 in cell (1, 2) allows us to cross out the 1's in cells (2, 2), (3, 2), and (1, 5).  Now we know that (3, 2) must hold a 6 and (2, 2) must hold an 8.  The 6 in (3, 2) lets us fill in a 1 in (3, 5) and a 9 in (3, 6).

The 3 in cell (1, 4) lets us eliminate the 3 from cell (1, 7), leaving a 2 there. Crossing out the 2 in cell (1, 5), only a 4 is left. Similarly, eliminating the 3 from cell (1, 7) leaves a 2 there.  That 2 can be used to solve cell (2, 8), leaving a 7 there.  That 7 allows us to solve cells (7, 8) and (8, 8), placing a 5 and a 6 in those cells, respectively. Now the 7 in cell (2, 8) and the 5 in cell (7, 8) together allow us to cross out everything but the 3 in cell (3, 8). The same 7 in cell (2, 8) also gives us the solution for cell (2, 9).

Crossing out 4's in box 4 lets us complete that box. Box 6 can be solved using the new values in box 4. Now every cell but one each in the upper- and lower-left boxes has been solved, and they can now be solved easily. Finally, we have come up with the solution.  It must be unique since every step we took was forced. However, the order in which we took the steps was arbitrary.

Fortunately, not every puzzle can be solved by elimination alone. Next time I'll show how the rule of uniqueness can be put into play.

 

Comments

[Kit Winterer]

I don't get how you get the 'one' in the upper left box. . . 'Continuing' just doesn't tell me how the '8' got eliminated for that cell.

[sudokublog]

Hi Kit -

I've added some more text to help clarify things. The 1 on the upper left box can be written in because we can simply eliminate all the other possibilities based on numbers in the same row, column, and box.

[Ed]

Hi Dan,

I'm trying to understand the logic you used that allowed you to place the 7 in the upper middle box after you placed the pencil marks. (3rd graphic) Will you elaborate for me please?

[sudokublog]

Hi Ed -

Let's look at that cell (cell (6,3), in the corner of the upper middle box). When we start the puzzle, we can see that it can't hold a 1,2,3,4,5,6 or 9, because all of those values exist in row 3, in column 6, or in other cells of the upper middle box. Only 7 and 8 are left as possible values.

Now look at cell (6, 6) (in the middle box). Just looking at row 6, column 6, and the other cells in the middle box, we can rule out every number except 8. So we know an 8 must go there.

Now if we revisit cell (6, 3), we can cross out 8 as a possibility since we have found the 8 in column 6. Since only 7 and 8 were possible values, we know that there must be a 7 there.

Hope this helps!

Dan

[eric]

You might want to backtrack and correct your spelling of sudokU in your first sentence, not sudokO. I know it's minor, but the same way a computer programmer might get picky about code, a Japanese speaker is picky about the usage of Japanese words... especially when so many people mispronounce it so many different ways.
I am enjoying the blog otherwise, keep up the good work!

[eric]

You might want to backtrack and correct your spelling of sudokU in your first sentence, not sudokO. I know it's minor, but the same way a computer programmer might get picky about code, a Japanese speaker is picky about the usage of Japanese words... especially when so many people mispronounce it so many different ways.
I am enjoying the blog otherwise, keep up the good work!

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The Vodafone pay as you phones offer flexibility and a wide range of options to make calling and communicating a pleasure ride.Factors that you will want:&bull Gentle brush&bull Lint no cost towel&bull New water&bull Gentle detergentHow to cleanse it:&bull Clear absent the filth and dust particles from the boots with the help of a gentle brush.Here comes the most essential piece-servicing. No wonder the ladies and the gals in the streets are all in UGGs.This type of footwear is also termed sheepskin boots as it is produced from lavish merino sheepskin. It can lead to a long lasting damage. This deluxe content not only has a incredible influence in wicking coldness absent but also helps avoid sweat from forming when the weather conditions will get warm. Here's it is offered in a easy way. As a best blend of UGG Traditional Quick and UGG Classic Cardy, UGG Bailey Button boots have develop into a favourite alternative amongst manner conscious people today all over the earth. You need to get it done on the frequent foundation to prevent your boots dropping its allure&bull Constantly continue to keep in thoughts that cleaning doesn't necessarily mean that you will place them inside a washing machine.&bull Dilute the detergent with drinking water and keep the solution aside.&bull Make confident that dust particles are removed before applying the answer.Current Affairs As you know that Ugg boots are popular for its top quality and ease and ease and comfort but this Ugg bailey boots is mix of style, glamour, good quality and definitely convenience. You must maintain your footwear and apparel to make them appearance great and enhance the longevity. Their sheepskin insoles sense so gentle and warm that you can put on these footwear not having socks.&bull Rub the fur gently and wash it with contemporary water. This winter season months include whole lot of colors, really like and happiness to your living instead of dullness and boredom. She was always working. It is possible to stay home with your kids and do something for yourself, all at the same time. Work-At-Home-Magazine isn't about wild dreams; it tells you truthfully how you may stay home with your kids and make some money while doing something you really enjoy."Women today are put into a difficult situation. Many women get up early, get the kids dressed, drop them off at day care, drive in rush-hour traffic, and arrive at work exhausted, thinking, "There has to be a better way.Choosing to work at home can be a tough financial decision."If we do decide to stay home with our kids, we're blamed for wasting our education. I love the Spanish proverb that says, "A rich child often sits in a poor mother's lap. Many leading fashion stylists advise their clients to purchase several pairs.. These unique boots are well known around the world. This boot has the uncanny ability to adapt into the changing trends within the fashion industry. They have even received a nod of approval from Oprah Winfrey. They are known to command attention and they also have the ability of going well with any outfit. Makers have never compromised the ease and comfort of these shoes over the years. Their fit, finish, feel, and comfort have remained the same over the years. Many women get up early, get the kids dressed, drop them off at day care, drive in rush-hour traffic, and arrive at work exhausted, thinking, "There has to be a better way. Working at home is a perfect solution. Work-At-Home-Magazine isn't about wild dreams; it tells you truthfully how you could stay home with your kids and make some money while doing something you really enjoy. "How hard can that be, really? She probably just sits in front of the television all day."If we do decide to stay home with our kids, we're blamed for wasting our education. She was always working.
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