PaulIQ164 wrote:Undoubtedly it is true that any solver program can fairly easily be modified to create puzzles (and the quality of the generator is defined by the quality of the solver). But in reality there's more difference than that, in that solver programs are less likely to give you features such as timers for your solving, graded difficulties, and most importantly, solver programs won't undercut the market of Pappocom, who host this forum advert-free and at no cost to us as users.

In the course of a puzzle we can usually find at least one move which has not been made yet but which must be made: a vacant square which can accommodate one and only one symbol or a row, column or block, one and only one of whose vacant squares can accommodate a certain symbol. The human solver finds one such move, makes it and looks for another. The machine solver finds all such moves and may as well count them. If the count remains high until the grid is nearly full the puzzle is easy because it is is easy for the human solver to find one such move if there are several. If the count drops to one or two while the grid is still sparsely populated the puzzle is difficult and Pappocom call it "fiendish". If the count is always positive we always know what to do and the puzzle is simple no matter how difficult it is. I wonder if there is a maximum difficulty simple puzzle, one whose count is always exactly one.

If the count drops to zero we are out of Pappocom territory and the puzzle is no longer simple so we might call it complex. The most we can say is (for instance) the 6 in row 3 must go in one of two squares, the 5 in column 6 into one of three....etc.; we grasp one of these forks and follow each of its prongs. Each prong leads (possibly via more forks) to a solution or to a dead end. It is easy to see that any fork will lead to all the solutions even though different forks lead to different collections of dead ends. By treating solutions and dead ends on the same footing we get all the solutions. If the puzzle is valid and so has just one solution we get a proof that this is so. If we want to know how complex a complex puzzle is we have a little more work to do: it is not quite enough to count dead ends; strictly we should try the effect of grasping different forks. Maybe the two placings of the 6 in row 3 both lead to further forks, giving one solution (and a proof of validity) and three dead ends while the three placings of the five all lead to simple situations, one soluble, two not, giving the unique solution with just two dead ends. A solver will always tell us whether a proposed puzzle is valid or not and, at no extra charge, whether it is simple or complex and, at almost no extra charge, whether it is easy or difficult.

A human solver working against the clock will sometimes treat a simple difficult puzzle as a complex one. This is what Vorderman did with both her published attempts at #89 from Wayne Gould's first book. You gallantly leap to her defence. I ungallantly maintain that a tournament chess player who failed to spot a mate in three in the middle game, went on to win the end-game and wrote it up for a column would present it as a blunder, not a triumph. The diagram would show the position where the mate was missed and the problem for the reader would be to do better at leisure than the writer did in the tournament.

I do not understand the comment about undercutting markets. I do not know who composes the puzzles published in books other than Wayne Gould's and newspapers other the The Times; it could be Pappocom or it could be a rival. If there are rivals out there let them undercut each other: that is what a market is.