by **nd** » Wed Nov 16, 2005 10:52 pm

Just a quick note before I knock off work. The first one was a learning experience: I started with a particular "key" I wanted (in the lower right corner) then went from there. I'm not sure how professional puzzle-creators go about their business: you can basically approach things from two directions:

1. write out a grid of numbers, then draw boxes around them & pencil in the sums; this guarantees a soluble puzzle but I can't see how you could really finesse the solving process this way (i.e. control the entrypoints & strategies for solving it); & it wouldn't necessarily be easy to produce a unique solution. (This is to my knowledge the approach used by computerized puzzle-creators: randomly fill a grid with numbers then draw random cages around them.)

2. write out the particular combinations of cages you want, carefully "solving" the puzzle at each stage as far as possible, then adding more, until the puzzle is finished. The problem with this strategy is that, as I said, it's very hard to avoid creating impossibilities, though I found that the 2nd time around I had a lot better idea of how to avoid this. It helps if you photocopy the half-finished puzzle at key stages, before you do something that may wreck it.

I suspect that professional creators use a combination of both methods. The other thing is that it's a lot easier to create puzzles if you avoid "all-over-the-place" design: for instance, in my creation of #2, I worked out the middle row of nonets first, then worked out the bottom row, then worked out the top row. (Unless I've screwed up, I think this is probably the simplest way to approach solving it too, beginning with the middle nonet.) In other words I was avoiding working both the horizontal and vertical axes at the same time, which is a way to quickly get into contradictions. That said, that strategy does limit the "kind" of puzzle you can create so I want to work out ways of doing all-over-the-place design which are less error-prone.

26 minutes, eh? That's about what I'd hoped for. Chris's posted solution to #1 is correct. I should add that, while there may be other ways in to the puzzle, the one I'd begun with was a plus-minus variant of the 45 rule in the bottom right nonet (the sum of the 3 cages is 20 + 11 + 6 = 37, so the difference between innie & outie must be 45 - 37 = 8, leaving 1 and 9 as the only possibilities). There's a slightly fancier variant of this idea in the 2nd puzzle.

The last few postings raise an interesting topic: in what situations is it possible to fix an erroneous solution? Is there any strategy that's not dependent on backtracking? (i.e. remembering your immediately previous moves & undoing them). (Thought experiment: someone hands you an unsolved puzzle where someone has made a mistake. So you don't know its past solving history. Is there anything salvagable or do you just have to erase it & start over?) -- I suspect that the answer to my questions is "almost never", & that knowledge of past history is necessary, but perhaps some of the more mathematically/computerly inclined might be able to give a firmer answer....