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