I'll leave the column switches to your imagination, but yes, that's right, I gave you essentially the same puzzle to solve twice. By the same, I mean isomorphic - which is just a fancy way of saying there a set of grid manipulations you can perform on one puzzle to get to the other. For the record, they are:
- Relabelling. That is, applying a permutation to the givens. There are 362,880 (=9!) of these.
- Dihedral symmetries - which is a fancy way of saying rotations and reflections of the square grid. There are 8 of these.
- Outer Box shuffles. Probably best highlighted by an example. Imagine taking rows 1, 2 and 3 from the top of a puzzle, and sticking them on the bottom of the puzzles so they become rows 7, 8 and 9. Any analogous operation which preserves the 3x3 bold grid defining the 3x3 boxes counts as an outer box shuffle. There are 36 (= (3!)^2 ) of these.
- Inner Box shuffles. Again I'll highlight this with an example. Imagine taking rows 1, 2 and 3 and shuffling them so they become rows 2, 3 and 1. Any analogous operation preserving a 3x9 or 9x3 section of the grid whose boundary is part of the 3x3 bold grid counts as an inner box shuffle. There are 46,656 (= (3!)^6 - count 'em!) of these.
Notice that you can get a horizontal or vertical reflection via a series of both types of box shuffles, as well as a 180 rotation, although you can't - thanks Thomas - get a 90 rotation. Put another way, there is a total of 1,218,998,108,160 (not counting any further symmetry within the partitions in the grid) things you could possibly do to turn one sudoku puzzle into a sneaky doppelganger. Those are the sorts of figures one normally associates with national debts....
Now, I'm not going to do any formal analysis on the times everyone kindly reported to me. I suppose for one thing I'd probably need in the order of 100's or 1,000's before being able to say anything worth saying. However, my theory was that since both puzzles were isomorphic, the second puzzle people solved would be slightly faster than the first. There ought to have been some sort of subconscious familiarity whilst solving, because combinatorially you could exactly the same thing to get to the solution!
I think it's fair to say that it didn't quite work out like that. Granted, I put a little effort into making the puzzles seem as different as possible, but the varying times seem to indicate that exactly how the grid is presented makes a difference as to how people have solved the puzzles.
I guess the other interesting thing is that many good puzzle solvers completely missed what was going on, despite the meta-signpost of an experiment with two rather innocuous puzzles. Perhaps some sorts of grids are more open to this sort of thing than others. Anyhow, I'm going to leave the story there for now, but perhaps you'll be hearing more on the subject from Thomas S or Grant F in future weeks.
This week's puzzle is something completely different - Yajilin. I made this hungover, and in rather a bad mood yesterday so don't be expecting too much, but do watch your step. enjoy!
#199 Yajilin – rated easyAll puzzles © Tom Collyer 2009-12.
Thanks for the interesting experiment even with limited sample size. I do have more to say on this subject but cannot predict how quickly I'll finish my recent analysis.
ReplyDeleteTo answer your question, you do need one dihedral element to explore all possible puzzles. To show necessity, consider the digit distribution which might be even (222222222) in the rows but unevenly distributed (123123123) in the columns. Certainly there are isomorphic puzzles that have an even distribution in the columns and not the rows, but shuffling will not permute these values, just rearrange them.
"However, my theory was that since both puzzles were isomorphic, the second puzzle people solved would be slightly faster than the first. There ought to have been some sort of subconscious familiarity whilst solving, because combinatorially you could exactly the same thing to get to the solution!"
ReplyDeleteI think there would probably have been better results with harder sudokus. Here we have easy ones, that can be solved only with hidden singles, so there isn't some specific technique or logical path that we could easily remember, consciously or unconsciously.
Fred, perhaps you are right, although that might prove to be something of a double-edged sword if the solving path is too narrow! If anything, the varying times raises fascinating questions about which particular solving path is taken based on the presentation on the grid... perhaps some people always start at the top left, or perhaps others go and scan for individual digits first.
ReplyDeletenevertheless, the end of the puzzle in both cases focuses on the centre 3x3 box. Perhaps people will look back and think about the different ways they got to that point of each puzzle!