Mathimagics wrote:I'm nearly ready to test your conjecture on uniqeness in completely specified Futoshiki grids.

It was only a question. I don't have enough data to give it the status of a conjecture.

I still have doubts about the "always" - which is not in itself a big problem if one gets uniqueness "often enough".

Mathimagics wrote:I've got a solver and a random Latin Square generator, just have to bolt them together.

I hadn't used my LS generator for some time - are you familiar with an elegant method by Pittenger? It's done by simple perturbations on an existing LS. It has genuine Markov chain properties, very close to truly random LS's.

I didn't work on Latin Squares

per se (as many people here, I was more interested in Sudoku). Of course, SudoRules can solve LS by switching off all mentions of blocks, but I tried only a few examples. The main reason is, I didn't have a LS generator.

AFAIK, what Pittenger leads to is a random (unbiased) generator of full grids. You still have to generate puzzles by a top-down process of clue elimination, which introduces some strong bias (wrt the number of clues), as in Sudoku.

For Sudoku, it's very hard to generate unbiased collections of minimal puzzles. Actually, as of now, we can't; we can only generate collections with (strong) known bias (see chapter 6 of my last book), starting from a list of all the non-equivalent complete grids (provided by gsf).

For Futoshiki, if uniqueness holds (often enough), one could try to combine a LS generator with the same "controlled-bias" techniques.