Hi

Cuorenero, Welcome on this forum,

Cuorenero wrote:Is there a sudoku 9x9 puzzle with a unique solution that can be solved only by making a guess at some point ?

All the known puzzles can be solved by singles after at most 3 guesses (but most of them after only 1 or 2); the minimal such number for a puzzle is called its backdoor size. The chances of making 3 correct random guesses are very low.

"Making a guess" must be distinguished from T&E (in Trial and Error, you just try a candidate z and eliminate it if it leads to a contradiction; you do nothing if it leads to a solution, i.e. you just keep z as a candidate and you forget this solution).

All the known puzzles can be solved using repeated T&E at depth 2 or less. This is a much stronger result than the previous one.

Still stronger: all the known puzzles can be solved with zt-braids(FP) for some relatively simple family FP of patterns (for details on all this, see the T&E vs braids" thread, here:

http://forum.enjoysudoku.com/viewtopic.php?t=6390)

Now, the real question is: what do you consider as an acceptable pattern? And the answer can only be subjective. If you don't accept nets, it is likely you'll have a very hard time with the hardest puzzles. But how many players want to solve EasterMonster & C° for breakfast?

From a theoretical POV, the way I see all this is: given some set of resolution rules, what can I hope to solve with them? Some answers are provided in the aforementioned thread for different families of braids. Of course, there may be some difference between what I can hope to solve with these rules and what I'm effectively able to solve (it is likely that a human player won't find all the possibilities of applying these rules in a grid). But I think this is still a valuable indication.

Cuorenero wrote:Another curiosity about all the known rules for candidate elimination: can they be reversed ? What i mean by that is: Can a candidate removal technique be used to place candidates in a bottom up construction of a puzzle ? Suppose you have a certain solution, one has to create a grid for which that filled square is a unique solution.

You remove a number, you place a candidate, or a set of them (according to general col-row-block uniqueness, or the aforementioned rules).

There's no

a priori reason why this couldn't be done - except perhaps the complexity of the procedure.

General purpose generators, such as suexg, generate random puzzles and can hardly provide very hard puzzles (which are very rare). For suexg, I've shown that all the puzzles generated in a collection of 1,000,000 could be solved by the simplest braids: nrczt-braids. (And for the first 10,000 of them, by simpler nrczt-whips.)

I've almost no experience in puzzle creation, but I tried to generate variants of EM with different x2y2-belts. I used SudokuExplainer. It has the advantage of propagating the elementary constraints. But it doesn't allow candidate elimination. It'd be a nice feature to be able to select a rule (or a set of rules) and apply all the eliminations it allows. That could help creating new puzzles in which no rule in the chosen set can be applied and which would accordingly be rated high.