by **Hammerite** » Mon Jun 20, 2005 10:31 pm

Thankyou scrose! And thanks also for the extra example...

Those examples show that 17 is an upper bound for k. I wonder, though, how much thought has been put into finding a lower bound... I will read some of the posts on this forum at some point.

I went for a stroll to Tesco and as I came back I had some thoughts... I considered what would happen if there were 7 clues in the grid. Clearly if there are 7 clues in the grid, then at most 7 of the 9 symbols will appear in the initial grid. Then at least 2 of the symbols do not appear in the initial grid. Clearly in any solution of the grid, the grid produced by the transposition of the 2 non-appearing symbols is also a solution. However, can it really be called a different solution? The two symbols are, by virtue of not being in the grid, invented by us, the solvers, and so if all that changes is that we transpose them, while keeping their respective locations the same, we may decide that the two solutions are in fact equivalent. If, however, having fixed one of each of those two symbols in one of the 18 cells those two symbols occupy overall, we can swap some or all of the symbols in the remaining 16 such cells and produce another solution, then the solution we had certainly isn't unique. If we can prove that we can do this in at least one way for any solution we come up with to any Su Doku puzzle with 7 clues, then we have shown that any Su Doku puzzle with 7 clues has either no solutions, or more than one solution, in which case a lower bound for k is 8. It would follow that k is 8, 9, 10, 11, 12, 13, 14, 15, 16 or 17. However, I would imagine that k is further towards the right of this list than the left.