123
456
789
You then shuffle columns 456 to get the digits "in order".
Same for columns 789.
Now swap, if necessary 456 with 789 so the r1c4=4.
Do the same with the rows so that r4c1=4.
You will obtain, for example
123 456 789
456 xxx xxx
789 xxx xxx
2xx xxx xxx
3xx xxx xxx
5xx xxx xxx
6xx xxx xxx
8xx xxx xxx
9xx xxx xxx
If we define Solution Class A,B as such a transformation where A is the values for r1c456 and B is the values for r456c1, then
Values for A (Solution class A,B)
456 457 458 459
467 468 469
478 479
489
Values for B (Solution class A,B)
235 236 238 239
256 258 259
268 269
289
This would give 100 solution classes.
Having picked a solution class, there are six cases of what goes into each of r2c456, r2c789, r3c456 and r3c789.
6*6*6*6 subclasses of Solution class A
It is more complicated in B, but probably the same number.
So total 12960x12960 = 167,961,600 Solution subclasses.
Lets look at a subclass of Solution Class (456, 235):
- Code: Select all
+---------+----------------------+---------------------+
| 1 2 3 | 4 5 6 | 7 8 9 |
| 4 5 6 | 9 8 7 | 2 1 3 |
| 7 8 9 | 3 1 2 | 6 4 5 |
+---------+----------------------+---------------------+
| 2 6 1 | 578 3479 34589 | 34589 3579 478 |
| 3 9 4 | 125678 267 158 | 158 2567 12678 |
| 5 7 8 | 126 23469 1349 | 1349 2369 1246 |
+---------+----------------------+---------------------+
| 6 3 7 | 1258 249 14589 | 14589 259 1248 |
| 8 4 2 | 1567 3679 1359 | 1359 35679 167 |
| 9 1 5 | 2678 23467 348 | 348 2367 24678 |
+---------+----------------------+---------------------+
Find the cell with the most candidates: r5c4, and check each one:
r5c4=1 gives 443 solutions
r5c4=2 gives 412 solutions
r5c4=5 gives 348 solutions
r5c4=6 gives 363 solutions
r5c4=7 gives 443 solutions
r5c4=8 gives 518 solutions
My guess is that on average, each Solution Subclass has about that many solutions: 2,500
This would made the total unique solutions in SuDoku
167,961,600 x 2,500 = 419,904,000,000
Rounding off: 420,000,000,000
Now if you take one of the unique solutions as defined above, you can reverse the process (re-assign the digit values at random, shuffle legally the rows, columns, and triplets). This looks like another solution, but it is just a restatement of the unique solution.
OK, now suppose we take each unique solution and create a puzzle. We have a stack of 420,000,000,000 puzzles to solve.
I've heard some guys here brag they can solve puzzles in, say, 3 minutes (on average). Assuming such guys can keep it up 8 hours a day for 300 days per year, it looks like about 900,000 years. Whew! Just wanted to let you know there is no fear of solving all possible puzzles in the near future.
Mac
