Sudoku and Kakuro


In recent years many people have played Sudoku, a number game in which a nine-by-nine grid is filled with the digits 1-9 so that each row, each column, and each group of nine squares contains exactly one of each digit.


(Puzzle drawn by Lawrence Leonard Gilbert, and released to the public domain.)

The game appears to have been invented by Howard Garnes, a contributor for Games Magazine: the puzzles began appearing there in 1979 under the title Number Place. They became popular in Japan and then, only a few years ago, in the United States. The puzzles lend themselves not just to recreation, but to mathematical thought: How many different grids are possible? What is the fewest number of squares that can be filled in initially but that still give rise to a unique solution? What is the greatest number of squares that can be filled in that don’t give rise to a unique solution?*

You can find more about the history of Sudoku in The New York Times article here; more mathematical information on Wolfram MathWorld here (including the little tidbit that Sudoku puzzles appears in an episode of Num3rs) or in the MAA Focus article “The Sudoku Epidemic” here. By looking around the web, too, you can find all sorts of variations on Sudoku: letters instead of numbers, smaller (4 by 4) or larger (16 by 16) grids, rectangular grids, overlapping grids, cubes….

Fans of Sudoku may also enjoy Kakuro, a number puzzle with a different twist. In Kakuro, the digits 1-9 are also used, but each row or column has to add up to a certain sum written above or to the left of the column. Within each sum a digit can be used only once, although not every digit is used in every sum. In the sample puzzle below, looking at the upper left-hand portion, the top row would have two distinct digits that add to 16, while the left-most column would have three distinct digits that add to 23.


In solving Kakuro it is helpful to keep track of unique combinations (for example, the only three distinct digits to add to 7 are 1, 2, and 4 in some order). Some people (like TwoPi) find that the extra challenge of Kakuro makes it more interesting than Sudoku; others (like Ξ) have yet to solve a puzzle without help and are primarily interested in finding the easiest versions. (What is the smallest possible puzzle with a unique solution?**)

*Answers: 6,670,903,752,021,072,936,960 different grids; 17 squares, many of which can be seen in G. Royle’s collection here; 77 squares

** Mark Huckvale hints here that it is 5 by 5, at least to be interesting.

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3 Responses to “Sudoku and Kakuro”

  1. GaussGirlRibby06 Says:

    I have played Sudoku a bit, and I have to agree with the New York Times article in that it is addicting. There is something intensely satisfying in working through the puzzle and filling in each box as you go. The context in which I am familiar with is the version on the Nintendo DS “Brain Age” game. This game is geared to get your mind working and activate your prefrontal cortex to improve memory and other mental capacities. I have noticed the more I play, the faster I get in solving each puzzle, and the more systematic I am about solving each one. This version of the game allows you to write possible answers in the corner of each box, and I find this to be extremely helpful. Like the article said, you must think about where a particular number can fit, but most of the puzzle is done by eliminating all the numbers that it could not possibly be. The harder the puzzles are, the less numbers or “givens” they give you, and they start to only give you only one or two 7’s or only one 6, so it becomes difficult to place that number anywhere in the grid until much later on when more of the board is filled. My technique is pretty basic: Start by looking for the most filled area, and I usually start with the number 1, and pick a row, column, or box and see if it can be narrowed down. I usually try 2 or 3 or whatever is next chronologically in that same area, and if not, I move to the next. The rule that I usually end up finding to be the most helpful is the number not being repeated in the 3 by 3 boxes. When trying to fit something into a row or column, ruling out 3 blocks usually yields a good result. It is amazing to think about all the possibilities of the puzzles that can be created, and that the least number of givens without having multiple solutions is only 17 (although it is technically not proven yet). They only need to provide you with 17 numbers, and you can fill out a grid of 81 squares! I think this game is so addicting to us because we, as humans, like to sort and file things and put them in their place, and it is an escape from the stressful things in the world (like “Cool Projects” for History of Mathematics class…) but without the “mind-numbing ness” of video games or television. Our brains like to be stimulated, and one could easily forget about the problems of the world for hours while being immersed in this logical puzzle. I think it is really fascinating, and many discussions on the mathematics and philosophy of these puzzles could be made.

  2. Batman Says:

    For those who enjoy sudoku and kakuro puzzles, I strongly recommend killer sudoku. Lots of puzzles can be found here.

  3. free sudoku Says:

    Nice blog, if you are into Sudoku you should give a look.

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