8. "Everest": A very simple solution to a very difficult Sudoku

In 2012, Arto Inkala released another very difficult Sudoku. He named it "Everest" because he felt it represented the very highest kind of challenge for solvers. It is truly a difficult puzzle. A while ago I was able to work out a solution to it after many hours of diligent effort. However, when I tried to re-create that solution for this channel, I found that I was not able to do so, even though I was following notes I had made about the approach I had taken! These notes were not a step-by-step record of the solution, they were a general description of strategy, and the puzzle was so hard that they were not sufficient to enable me to solve it again.

So I took a step back and thought about why the puzzle was so hard. I kept remarking how "stingy" it was. It seemed that no matter what I tried, I didn't get much for my efforts. Then I realized that this was the key both to the puzzle's difficulty and to its solution. It was designed to thwart solvers by leaving multiple avenues open. As one report about Everest explained when it was first published, "Most moves will leave you with two or more spaces where a number could conceivably go." But the puzzle nevertheless had a solution. And so somehow, I realized, a solver needed to get the puzzle to yield as much information as possible.

It dawned on me that a good approach would be to find places in the puzzle where the placement of a given number would reproduce as many more of that number as possible in other blocks. That was a simple principle to follow. After a bit of experimentation, I realized that placing a 1 where shown in the video would locate 1s in a unique cell (or, in a couple of cases, in one of two cells) in all of the other blocks. That would also create an "only two cells" situation for the 8 in one of the blocks, allowing an 8 to be placed in another block (as shown in the video) in such a way as to place 8s in unique cells in all of the blocks. One of those 8s also placed a 5 in a unique cell, and that allowed the placement of a 5 in another block (as shown in the video) in such a way as to locate 5s in a unique cell (or, in a couple of cases, in one of two cells) in all the other blocks.

And that, it turned out, was sufficient to solve the puzzle. Admittedly this is an unusual approach. It would probably not work for too many other Sudokus. But the unique design of this one meant that the principle I identified—placing numbers provisionally where they would reproduce themselves around the board to the greatest extent—led to a solution.

Update: Since solving this puzzle this way, I have been having good success solving Sudokus by starting with the number best represented and seeing all the ways it can be supplied in each block. Typically there are only a few ways, sometimes only two. I set up one way and then see how the next-best-represented number or numbers can fit in each block; if they don't fit in any arrangement, this falsifies the way I have set up the first number, so I try it another way and test the further numbers again. This typically leads to a faster, cleaner, and more enjoyable path to a solution than other methods I have tried in the past. I can see that this is basically the approach I took to Everest: Seeing how a given number could be placed in as many as possible of the blocks at once was one means of setting it up for the kind of testing I described. It just so happened that in the case of Everest this means led directly to a solution. But I can now see that it was a specific instance of a general principle.