40. Using “invisible numbers” to solve a Sudoku: Patience revisited
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2 ماه پیش
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The Sudoku called Patience had
The Sudoku called Patience had been for me “the one that got away.” I felt that I had found an interesting solution for it (video #22), one that demonstrated some valuable things. The solution was even inspired, in part, by a dream. However, I also felt that I hadn’t quite “gotten inside” Patience as I had for the other Sudokus for which there are videos on this channel. That is, for those other puzzles, I felt that I had understood the structure and design of each one and found a solution path suited to the character of each one. But I didn’t have that same feeling about Patience. So it was always in the back of my mind as a puzzle I’d like to take another crack at if I could find a fresh way to approach it.
My recent explorations with transposition provided that fresh way. I decided to see what would happen if I moved things around substantially enough in Patience that I wouldn’t recognize and be tempted to follow the lines I had taken in the previous solution (as had happened to me on a couple of earlier occasions when I’d tried to revisit it). So I rearranged several rows and columns and even flipped the two sides of the puzzle along a corner-to-corner diagonal. (I had just learned, from Eline Hoexum’s thesis [see video #39], that this was another valid means of transposition.) I thought it would also be good to move around some blocks. I noticed that one block was empty and said to myself, “I’m going to put that one in the center. It’s probably the key to the whole puzzle.” So I did.
And then it struck me: the empty block was the key to the whole puzzle. I saw that all the numbers except the 3 and 9 ran into it in such a way that they could each go only in various smaller portions of the block. These were, in effect, “invisible numbers” in that empty block. Their exact positions were not certain, but there was a good general idea of where they belonged.
The 3 and 9, for their part, could be put in places where they would rule out enough of the limited options for the other numbers that this would determine the placements of several of them. Then the solution might be on its way. While there were theoretically 72 different ways in which the 3 and 9 could be placed within the nine cells of the block, most of those were not of interest, since they did not create definitive placements for other numbers. The handful of locations that did start the ball rolling could be tested manageably. (I will admit that I actually got the locations for the 3 and 9 from my previous solution to Patience. I figured I’d earned that. But I’m sure they could also have been determined by a reasonable amount of testing; that in itself would have been enjoyable and interesting.)
As the video shows, the rest of the numbers can indeed be derived from this opening. And this begins in the empty block. Five of the first eight cells to be solved are in that block. The invisible numbers become visible and point the way forward.
This solution path follows the same “script” as for other complex Sudokus whose designs I believe I have understood and followed. As the numbers fill in, there is increasing complexity until everything reaches a “point of culmination.” Some special tactics are then required to break through an impasse, but after that, the numbers all fill in, one after another.
While this solution seems harmonious with the character of the puzzle, the tactics required are nevertheless quite involved in some places, requiring considerable deductive logic. This may help explain the name Patience. I said in my comments to the previous video on this puzzle that it was well named, in that it took a long time to solve. But now I saw that the name fit the puzzle in another way: in several instances, the case for a single number’s placement has to be built up slowly, over a series of steps. So patience is required to work out those arrangements. While there is not always (or even often) a connection between a Sudoku’s name and its character, for this one, perhaps there is.
For purposes of comparison, in this video I re-enact in the original layout of Patience the solution I was able to find in the transposed version. But I show the transposed version at the end. I think this experience shows that transposing a Sudoku can indeed enable us to see things that we would not see otherwise. In this case, it enabled me to see some numbers that were invisible.
Chapters:
00:00 Making invisible numbers appear
00:38 Some tactical fireworks
01:07 More tactical fireworks
01:33 Yet more tactical fireworks
03:06 A radical transposition
My recent explorations with transposition provided that fresh way. I decided to see what would happen if I moved things around substantially enough in Patience that I wouldn’t recognize and be tempted to follow the lines I had taken in the previous solution (as had happened to me on a couple of earlier occasions when I’d tried to revisit it). So I rearranged several rows and columns and even flipped the two sides of the puzzle along a corner-to-corner diagonal. (I had just learned, from Eline Hoexum’s thesis [see video #39], that this was another valid means of transposition.) I thought it would also be good to move around some blocks. I noticed that one block was empty and said to myself, “I’m going to put that one in the center. It’s probably the key to the whole puzzle.” So I did.
And then it struck me: the empty block was the key to the whole puzzle. I saw that all the numbers except the 3 and 9 ran into it in such a way that they could each go only in various smaller portions of the block. These were, in effect, “invisible numbers” in that empty block. Their exact positions were not certain, but there was a good general idea of where they belonged.
The 3 and 9, for their part, could be put in places where they would rule out enough of the limited options for the other numbers that this would determine the placements of several of them. Then the solution might be on its way. While there were theoretically 72 different ways in which the 3 and 9 could be placed within the nine cells of the block, most of those were not of interest, since they did not create definitive placements for other numbers. The handful of locations that did start the ball rolling could be tested manageably. (I will admit that I actually got the locations for the 3 and 9 from my previous solution to Patience. I figured I’d earned that. But I’m sure they could also have been determined by a reasonable amount of testing; that in itself would have been enjoyable and interesting.)
As the video shows, the rest of the numbers can indeed be derived from this opening. And this begins in the empty block. Five of the first eight cells to be solved are in that block. The invisible numbers become visible and point the way forward.
This solution path follows the same “script” as for other complex Sudokus whose designs I believe I have understood and followed. As the numbers fill in, there is increasing complexity until everything reaches a “point of culmination.” Some special tactics are then required to break through an impasse, but after that, the numbers all fill in, one after another.
While this solution seems harmonious with the character of the puzzle, the tactics required are nevertheless quite involved in some places, requiring considerable deductive logic. This may help explain the name Patience. I said in my comments to the previous video on this puzzle that it was well named, in that it took a long time to solve. But now I saw that the name fit the puzzle in another way: in several instances, the case for a single number’s placement has to be built up slowly, over a series of steps. So patience is required to work out those arrangements. While there is not always (or even often) a connection between a Sudoku’s name and its character, for this one, perhaps there is.
For purposes of comparison, in this video I re-enact in the original layout of Patience the solution I was able to find in the transposed version. But I show the transposed version at the end. I think this experience shows that transposing a Sudoku can indeed enable us to see things that we would not see otherwise. In this case, it enabled me to see some numbers that were invisible.
Chapters:
00:00 Making invisible numbers appear
00:38 Some tactical fireworks
01:07 More tactical fireworks
01:33 Yet more tactical fireworks
03:06 A radical transposition
2 ماه پیش
در تاریخ 1403/03/27 منتشر شده
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