So first things first - last week's experiment. Perhaps all will become clear if I permute [in disjoint cycle notation: (163754)(29)] the labels of the givens of the second puzzle, and rotate the grid.
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 easy
All puzzles © Tom Collyer 2009-12.
