Sudoku: Is It Winnable?

Many people that I have come into contact with believe that Sudoku is not winnable, well it is. I have solved many Sudoku puzzles over the past couple of years. It’s not that Sudoku seems unwinnable; it may just be that many people do not have the patience to find strategies to solve the puzzles. That used to be me. A couple of years ago I was determined to learn to play Sudoku. Yes, it was hard at first and very time consuming, but I eventually got the hang of it. I actually use it as a stress reliever now that I am in college. When I need to take a break from homework or studying, I pull out on of my Sudoku books and complete a puzzle. I am able to solve most puzzles in under twelve minutes. For anyone interested in Sudoku, I suggest that they do not try to solve a puzzle in one day. I suggest taking it slowly and solving bits of the puzzle over the duration of multiple days, this way they will not become overwhelmed and give up all together.

Sudoku requires no calculation or arithmetic skills. It is essentially a game of placing numbers in squares, using very simple rules of logic and deduction. It can be played by children and adults and the rules are simple to learn. The objective of the game is to fill all the blank squares in a game with the correct numbers. There are three very simple constraints to follow. In a nine by nine square Sudoku game: every row of nine numbers must include all digits one through nine in any order, every column of nine numbers must include all digits one through nine in any order, and every three by three subsection of the nine by nine square must include all digits one through nine. Similarly, smaller Sudoku puzzles, such as the four by four puzzles, must have the numerals one through four in each row, column and subsection.

Every Sudoku games begin with a number of squares already filled in, and the difficulty of each game is largely a function of how many squares are filled in. The more squares that are known, the easier it is to figure out which numbers go in the open squares. As you fill in squares correctly, options for the remaining squares are narrowed and it becomes easier to fill them in. The Sudoku games on SudokuDaily.net let you check your progress as you go, to help prevent going down a wrong path.

From playing Sudoku many times I have learned a few strategies to help me solve the puzzle. First scan the rows and columns to see where a certain number might go, given the three constraints listed above. For example, the fact that a seven is required in the top right corner can be determined by first analyzing its nine square sub-regions. The only numbers missing in the region are a five and a seven. However, putting a five in the top right box would conflict with the five already in the top row and the rightmost column. The seven, on the other hand, would not conflict with any of the given numbers. Once the seven is filled in, deduction requires that only a five can go beneath it as all digits from one through nine must be represented in the region. From there, one can turn to the two remaining open boxes in the right column – these must include a two and a four as the column’s digits must represent one through nine. One of these options, placing the four beneath the one, would lead to a conflict with the four already in that horizontal row, so the only option for this box must be a two.

However, options for boxes are often not that easy to deduce. Another technique is to “pencil in” possibilities and then follow the possible solutions that emerge until a conflict is found. Often these conflicts appear after two or three numbers are penciled in, and one can return to the start and try the next option until something clicks.

So, Sudoku is indeed winnable, you just have to be willing to put the time and effort into this unique set of puzzles.

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5 comments
  1. ultrapoulet said:

    Klondike solitaire is another game that some feel can be impossible to win. In some games, this is true. Some games of Klondike actually are impossible. However, most games are possible to win. The problem is, unlike Sudoku, which has strategies that will always work, Klondike solitaire lacks this. Strategies that might work in some games might not in others. From what I’ve seen, people have been unable to even determine the total number of games of Klondike that are solvable, let alone a universal strategy. On the other hand, people have solved for the total number of Sudoku puzzles, as well as the minimum number of initial squares need to be filled in. On top of this, there are universal strategies that can be used on most Sudoku games. If a game doesn’t have a real strategy and winning mainly boils down to luck, most people will feel the game to be next to impossible to win. Sudoku is hard, but does not feel impossible. Klondike on the other hand, does feel impossible at times, especially after a losing streak of ten games.

  2. So called “casual” games like Sudoku or Klondike solitaire are always the ones that make me think about strategy the most. That probably comes from playing so often. These strategies never come about from trying to develop a strategy specifically, but seem to come about from a more basic pattern recognition ability in the mind. Once that pattern recognition drive is sated and the game seems too easy, I move on. I guess it’s because once the patterns are recognized, the game feels mastered and lacks challenge. The brain is always seeking out new patterns to recognize.

  3. I agree with Nadeem. These kinds of games definitely are the ones that make me think of strategy the most. That’s often because there is not a lot of extra content to distract you. The game has very straightforward rules and expectations of play, and that’s really it. It seems like that simplicity could get quite tiresome, but I find that games like sudoku or minesweeper or freecell or tetris have the most replay value. Games that are so stripped down make it less about the game and more about the player.
    I’ve probably logged more time playing Tetris than any other game. I went 5 hour car rides as a kid doing nothing but playing Tetris DX on my Gameboy Color. I only had a few games, the others being Pokemons Red and Blue and Link’s Awakening, and I never got around to beating the others because I was so engrossed in trying to get a new high score. Tetris became less of a competition with the game and more of competition with myself. Would the iteration of me on US-23 near Saginaw overcome the high score set by the iteration of me that was getting on the Beck Road entrance ramp to I-696? What strategical improvements would Saginaw me make that Wixom me didn’t see? I suppose this kind of mentality can be applied to other games with speed-runs. It’s less about beating the games than perfecting your strategy running through it, but I don’t have the patience for that. I guess I’m more casual. I’m fine with that.

  4. Games, like Sudoku and Hex 7, are definitely strategy games. They make you look for patterns and they are extremely addicting.They keep the nervous, primate part of our minds that are always looking for problems and ways of solving them busy. These games keep our minds occupied and gives us the satisfaction of doing something without feeling anxiety or stress. I remember reading once that Sudoku puzzles were banned on certain TV production sites as well as stores. Their employees spent so much of their time on the puzzle that it was interfering with their work significantly. Games like this are easy to obsess over. I remember in high school after AP exams, all we would do is Sudoku puzzles. At first we were like these are impossible, but they weren’t. It just needed practice, as all strategy games require. We would meet together after school in the library just to figure them out and all we wanted to do was that. Some of forgot about our other classes and focused solely on the game. It reminded me of the obsession we had as kids with the Rubik cube. Where we would think solving it was impossible, but after great focus, effort, and intense thinking, it was solved, and once you solve it, you get this huge rush of excitement and feeling of accomplishment. This results in repetition and as I said earlier, obsession. The feeling of solving something that you thought was impossible, like a really intense math question, is probably one of the greatest feelings you will ever have and this feeling is something you definitely want to recreate.

  5. amiesi said:

    The math club just had a talk from one of the EECS faculty about algorithms of solving Sudoku puzzles. As she said, Sudoku puzzles are examples of Constraint Satisfaction Problems that usually show up in computer science. There are some really cool ways of solving Sudoku puzzles, such as the method of Exact Covers that usually can solve any 9×9 Sudoku grid instantaneously. But it should be noted that the complexity of the algorithm forces its efficiency to decay to unacceptable levels past a certain n number of rows and columns. And There are also algorithms of counting Sudoku solutions, and determining the number of possible Sudoku solutions possible for any given amount or number of numbers. I think the coolest part of the math behind Sudoku is the the creation of “impossible” Sudoku problems. These are actually not impossible, just literally beyond the scope of human logic, a notion which I find moderately amusing. The capability of Sudoku to create problems that are beyond the scope of human logic, but still within the scope of human imagination, comments on the versatility on the human hardware but the fragility of the human software.

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