This Sudoku has a pattern completely unlike the ones I’d become familiar with by working on the set of ultra-difficult puzzles that a group of theorists created a few years back. In my previous video, on Silver Plate (#43), I described my approach of seeking to recognize the design of a Sudoku and the relationships between its “givens” and then putting in numbers that would reasonably start things down the path to a solution. Admittedly it was easier for me to recognize the pattern in Silver Plate, since I’d already seen a similar pattern in several other Sudokus. I wondered how successfully I could follow this approach with a Sudoku that had a significantly different pattern.
Just then I came across an article by Daniel Beer, a software engineer in New Zealand, entitled “Generating difficult Sudoku puzzles quickly” (dlbeer.co.nz/articles/sudoku..... In that article, he described his algorithms and provided several illustrations of the Sudokus of varying degrees of difficulty that they produced. He rated his puzzles on a scale of 0-999; only about 1% of them scored above 900. One of his illustration puzzles was rated 953.
So this seemed like the perfect test for my approach. The computer algorithm wasn’t “aware” of any kind of pattern at all. It was simply ensuring that the Sudoku had one unique solution and that it could be reached from the givens. Nevertheless, if there was a solution, there had to be a path to it from the givens, and so there was presumably some kind of pattern in the numbers that reflected this path.
As I looked over the puzzle as a whole, I noticed that the second and eighth rows were each missing only three numbers; that two of these numbers were the same for each row (1 and 5); and that two empty cells in the second row were in the same columns as empty cells in the eighth row. I realized that this meant that the key to the puzzle was reasonably the location of the 1 in the eighth row. So I did some testing to determine its location. Because of the connection between the second and eighth rows, wrong avenues were quickly falsified. It turned out the 1 went in the eighth column in the eighth row. This further placed the 5 and 9 in that row, and since the 1 was in one of the columns where there was also an empty cell in the second row, this determined the placement of the 1 in that row (since it had to go in one of those columns). And so things were on their way. The first 1 I had placed ultimately turned out to be a “magic square” (see video #13) that permitted all of the other cells in the puzzle to be filled in. Some involved tactics were required in places, and a forcing chain was needed at one point, but I felt that this experience was a positive one for the approach I follow. (After I solved this puzzle, I ran it through the Sudoku solver at sudokuwiki.org, and it rated it “extreme.” So this was an appropriate challenge.)
By the way, if you’ve ever wondered how computer algorithms generate Sudokus, I recommend Daniel Beer’s article. It is very helpful and clear.
I was intrigued by the thought that a human designer trying to create a difficult puzzle probably wouldn’t make one that allowed solvers to find the first seven numbers easily, as happens here. But the computer, without taking such things into consideration, simply determined that getting from the start to the solution in this puzzle would be more difficult than in 99% of other puzzles. And that was true even with the easy finds at the start. The algorithm wasn’t “trying” to create a difficult puzzle; it’s just that when computers generate a lot of puzzles through random processes, some will be more difficult than others. So purely computer-generated Sudokus offer an interesting test case of the idea that “if there’s a solution, there’s a path to it, represented in a pattern.” I may explore this further.
Chapters
00:00 Setting the stage
00:22 A “magic square”
00:37 Some involved tactics
01:03 More involved tactics
01:29 A forcing chain
02:11 Yet more involved tactics
19 июл 2024
