by **Guest** » Mon May 23, 2005 6:28 am

I was doing the Medium puzzle in Perth's Sunday Times (22/5) and made a mistake somewhere early on. I overwrote one of the pre-printed digits. The interesting thing is that I managed to come up with a solution based on that wrong digit. I also discovered that there was an alternative solution again based on that wrong digit. Both solutions conform fully to the rules of Sukodu, so I believe that it is possible to have more than one solution. These are the 2 alternative answers:

694 823 157 * 684 923 157

287 951 643 * 297 851 643

531 467 289 * 531 467 289

173 248 965 * 173 248 965

465 139 728 * 465 139 728

928 576 431 * 928 576 431

346 715 892 * 346 715 892

759 682 314 * 759 682 314

812 394 576 * 812 394 576

The difference between the two is that the 8s and 9s in r1c2, r2c2, r1c4 and r2c4 are transposed (8 becomes 9 and 9 becomes 8). It is obvious why both solutions are valid. The only items affected by the transposition are row 1, row 2, col 1, col2, mini-grid 1 and mini-grid 2. The transposition has the same affect on all of these, in that where there was an 8 there is a 9 and vice-versa. So by definition, each of these still contain all nine digits.

One way of making either of these a 'unique' solution is for the problem setter to pre-set any one of these 4 cells to either an 8 or a 9. But if all are empty to begin with (as was the case here), then more than one solution is possible.

In fact with careful planning I suspect that one could compose a problem with not just 2 but possibly many solutions, if it contains multiple cell-combos like that shown above.