July 17, 2020
Updated on July 18, 2020 at 22h51
The”art” of sudokus, it is also the art quote !
SCIENCE DAILY / “A question I turlupine for a long time… I guess that the sudoku grids are constructed by computer, but how do we determine in advance the level of difficulty of a grid ?”, request Michelle Caron, of Quebec.
There is no single way, or even universally shared, to determine the level of difficulty of a sudoku. In some cases, it is “measured” by simply counting the number of boxes in which a digit is found from the start : the more squares already filled in, the more it is considered that the puzzle is easy to solve. It is true that, in general, give more clues makes the task easier, but sticking to this is really only a very, very rough to the complexity of the grid.
Because of this, many generators of sudokus have developed ways much more elaborate to note the difficulty of their puzzles. The scales and scoring systems vary from one company to another, but in consulting with a few, one quickly realizes that the same basic principles of constant — see, in particular, about the site sudokuoftheday.com or this text from the British, Andrew C. Stuart, who manufactures him as grids and who has written books on the”art” of sudoku.
Among these principles, the most fundamental is without a doubt the level of complexity of the techniques needed to complete the grid. The most basic is to simply use the spaces filled to find someone, or empty boxes, for which there is only a single number is possible (or a single “candidate”, as is sometimes said). A sudoku that can be solved entirely in this way is obviously very easy — and incredibly annoying, I might add, but that is another question.
Other puzzles require the use of techniques a little more developed, such as the “double-hidden”, that is to say, when two boxes of the same row or column can only contain a pair of numbers-candidates. It is not known which digit goes in which box, but the fact of knowing that they occupy nécessaiurement these two boxes (no matter in what order) is already a hint of more.
And there are also methods even more complex, such as the technique known as swordfish. Roughly speaking, this occurs when a single “candidate” appears in two empty boxes of three columns AND three rows (this gives 6 boxes in all). If we can detect this pattern, then it is known that the candidate will necessarily be present in three of these six spaces, and nowhere else in these three columns/rows. This eliminates other possibilities and continue to make progress.
Once again, every “provider” of sudoku will rate the difficulty of these techniques differently. Some will take into account the fact that a technique must be used to get out of a cul-de-sac, the number of times that each method must be deployed, etc, The results may vary but, in general, the results are approximately compatible.
Interestingly, Andrew C. Stuart went so far as to measure the time taken to complete its sudokus according to their difficulty (as determined by his scale with him). And it seems that the way to measure the complexity of a sudoku is quite valid, since its grids are “easy” to take around 15 and 20 minutes to complete, its “medium-sized” between 19 and 28 minutes, its “difficult” between 25 and 45 minutes, and its “evil” between 35 and 50 minutes.
Another item to mention : Mr. Stuart has compiled the statistics on several thousands of sudokus. Thousands ! And why was he given all this evil ? “We all have different talents and to different degrees, so that some puzzles will seem easy to some and difficult to others, he wrote in 2007 in a text called Sudoku Creation and Grading. (…) Well rate the difficulty of a sudoku is the main concern of any maker of a jigsaw puzzle. If too many people disagree with your rating, then you will most likely lose your audience.”
Hence the importance, for him, to do what it takes — regardless of the amount of work this represents — to note its grids.
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