7. Setting the numbers in motion (Arto Inkala 2010 puzzle)

Arto Inkala released another puzzle in 2010 that he said was even harder than his 2006 puzzle “Escargot.” I certainly found that to be true. Once again I spent many challenging but enjoyable hours working on this Sudoku. And at one point, it felt as if the the numbers had been set in motion so that they marched through the puzzle to a solution. Here's how that happened.

I was interested to note that as a result of the initial cross-hatching, there were three instances of “only two cells” in the upper-left block. Two of them were end-to-end. (This is illustrated in the video.) As I pursued further avenues towards a solution, I noticed that if I put a 6 in row 4, column 3 (r4c3), this created an “only two cells” situation in the upper-left block that connected the other three instances there. And as I pursued some further avenues, I noted that putting a 2 in r5c1 created another “only two cells” situation at the other end of this chain. In fact, with the 2 and the 6 in both of those positions, “pushing” on the whole chain from either end placed a number in every cell in that upper-left block. That seemed worth pursuing.

This might not lead to a solution in the case of many Sudokus. But I knew that this one had been carefully crafted by a human being to be very difficult to solve but at the same time to have a solution. I thought that perhaps I was seeing the framework of that solution. So I tried “pushing” numbers into the chain from either end. A push from one end (the 1 from r2c3 to r1c1) turned out to provide enough information to solve the whole puzzle. One thing that helped was that several numbers in the upper-left and middle-left blocks had counterparts on the right-hand side of the puzzle or in the lower-left block, and so the cells filled in readily. That was what made it feel as if the numbers had been set in motion. This gave me confidence that I was indeed seeing the design of this Sudoku, and in more general terms, it made me think that in the case of many puzzles, there may be a path built into the puzzle that will set the number in motion to a solution.

Now someone may say, “Sure, let me add a couple of numbers to the original ones, and of course it will yield a solution.” Granted. But the challenge is to know what those numbers are and where to put them. I think that figuring that out is another aspect of solving Sudokus that deserves our attention. It is, in effect, just as much a “technique” as the ones that we know by various colorful names and that we use to make progress in puzzles. Several of those techniques are involved in the solution here.