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14. Platinum Blonde: A Sudoku’s constraints shape its solution

I recently learned of a very difficult Sudoku puzzle called Platinum Blonde. It dates back to at least 2009,* but people are still discussing it and talking about ways to solve it. It is one of the most difficult puzzles I have ever seen. An article by I-D. and A-I. Nicolae, “Limiting Backtracking in Fast Sudoku Solvers,” says: “As a general rule, the fewer clues are given, the harder the puzzle is, but this is not universally true. One of the most famous exceptions for this rule is the so-called ‘Platinum Blonde’ grid, where the apparently comfortable number of initial clues, 21, is in contradiction with its ultra-hard level of difficulty.” I spent some time getting acquainted with this puzzle, pursuing various directions through it, and eventually I recognized its internal logic. I then spent an enjoyable time working out a solution. I found the puzzle so elegantly structured that I wanted the solving experience just to keep going. I almost didn’t want to reach a solution. Well, al-most. I did solve it, and this video shows how. (See link at bottom for a second solution. The experience did keep going!) For one thing, as illustrated (0:12), the 5s and 8s provide a structure for the puzzle. There are three numbers in row 3 (a key row for the solution), and each one has a 5 and an 8 below it. Also, most of the binary options that can be identified with Snyder Notation are either 5s or 8s. However, the 5s and 8s do not figure centrally in the progress to the solution. Rather, they seem to provide a constraint within which that progress must take place. They provide a container for the solution, rather than the solution itself. There are only three other Snyder Notation pairs (0:16), and I found that they function like “switches” for the solution, in the sense of railroad switches that guide trains onto the right tracks. The right setting of the 9 (0:33) and the 6 (1:15) keep the solution on track, and the 4 (0:50) similarly plays an important role in constraining the solution to one direction rather than another. Because of the constraints on the 5 and the 8 in the upper right-hand block (0:18–0:24), two cells in that block have to be either 67, 69, or 79. Choosing two numbers for those cells will always determine the placement of those numbers in row 3. That is illustrated with the 9 and 7 at 0:44 and 1:04. Things work similarly with the other possible choices, but with the result that they are quickly falsified. No matter what the choice, in other words, the puzzle showcases its elegant structure. The solution emerges from the pathway shown in the video. After some hard-earned numbers, there is a breakthrough and the seven missing 2s fill in sequentially. After that, the remaining numbers also fill in, one after another. In an article that uses Platinum Blonde as a test case, “The Chaos Within Sudoku,” Maria Ercsey-Ravasz and Zoltan Toroczkai write, “The mathematical structure of Sudoku puzzles is akin to hard constraint satisfaction problems.” I feel that the approach shown in this video identifies the constraints within which a solution must lie and then finds the solution within those constraints. Admittedly this requires supplying numbers. This is a different undertaking than looking for an algorithm that can solve Sudokus of any level of difficulty linearly, without backtracking and without supplying numbers. I wish those who are pursuing such a goal every success. My own goals in doing Sudokus are relaxation and enjoyment, and this one certainly served those purposes very well! I hope you enjoyed it too. Thanks for watching. Update: After I got a "flash of insight" while working on tarx0134 (video #18), I wondered what would happen if I applied the same insight to Platinum Blonde. Frankly, I was amazed. Have a look at the results at    • Platinum Blonde revisited: A fascinat...  . *Most sources cite as a reference for Platinum Blonde a January 9, 2009 post to the Sudoku Players’ Forum, “The hardest sudokus.” There it is credited to a Sudoku designer and theorist who went by the name of eleven on player forums. A tip of the hat to the creator of this wonderful puzzle.