Mathimagics wrote: I somehow got hooked up to the site linked at the top of this thread! [...]But it does look like he is definitely doing generation, as I described ...
I agree that kakuro-online does online generation. I'ven't tried many of their puzzles but all the "hard" ones I tried were terribly easy and boring.
Mathimagics wrote:You remember that the original idea was to demonstrate the unlikelihood of getting a useful Kakuro puzzle out of a random generator, ie. by generating a valid grid at random and then calculating the block sums, hoping to stumble across one that is well-formed, ie unique for those blocksums
The only way I could easily do that was to take an NxN grid, all cells being white, with "vsums" being the sum of each column, and "hsum" the sum of each row. I generate all the valid squares, write them to a file and then sort them by the row/column sums. I then load the sorted file and only then can I tell which sum combinations are unique, and thus get a reasonable estimate of the probability of a random square being a unique combination, by counting the number of sum combinations that were unique.
OK, now I understand. But I don't know if it allows any conclusion about real Kakuros with inner black cells.
Sticking to my idea of density of black cells as a main parameter for the probability of having a 1-sol puzzle, and considering the propagation bar in the Kakuro-online generator, I suppose what they are doing for their "random" generator is the following:
1) choose size
2) choose some pattern of inner black cells (necessary for large size), maybe from a fixed database of patterns
3) fill in a complete Kakuro grid with this pattern
OR: 2+3) choose some complete Kakuro grid from a fixed database
4) compute the sums
5) delete the white givens one by one until just before multi-sol (maybe with some backtracking in order to delete more givens)
6) for each remaining white given, add an inner black cell (replacing this given), modify and complete the sums accordingly - this would be when you see the bar fluctuating, near the end of generation; the real process may be different, but black cells are somehow added, unless they start the whole thing with only high density patterns
7) if it doesn't work (i.e. can't delete all the white givens without loosing uniqueness), restart from 5
Simple, but the result is a very high density of black cells and easy puzzles