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Number: Puzzle #413
Genre: Panorama
Author: Maarten
Appeared at: September 26, 2007
This one was really nice!

I never really got the hang of solving Panoramas. I solved a few 1* and 2* ones in the past and that was a struggle. Yesterday I decided to make a serious effort to 'crack' the genre and became so engrossed in them I solved over a dozen, spending entirely too much time doing so.

Thanks! I must say I like the dynamics of Panoramas, and I had a lot of fun making those puzzles. It's good to hear somebody enjoyed them. (And that I'm not the only one spending too much time on them.

)

By the way, I thought I'd let you know the method I thought up to solve them. I wonder if it's anything like how you 'intend' them to be solved, because it doesn't really match the way the example puzzle's logic comes across.

I don't think these should be considered spoilers but I'd rather err on the side of caution, so...

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I use 2 colors: blue for hexes with confirmed heights, red for hexes with as of yet uncertain heights.

I start out by painting every hex with the greatest height it could possibly have: a blue shoreline (always a certainty) bordered on the inside by a red ring one level above shore level, iteratively another level higher within that, until the whole island is basically one big (usually symmetrical) 'mountain'.

Then I start cutting the island down to size. Red hexes at higher altitudes often 'see too much' of the island, and because all hexes are at their highest possible level, that can only be fixed by reducing their altitude. A change like that cascades into surrounding hexes because every hex must remain within one level of height difference with its neighbors.

Whenever a situation arises where it becomes clear (through all kinds of logical considerations that are too numerous to go into here) that a hex cannot have a lower altitude than I currently have it at, I change its color to blue. The two most obvious reasons to paint them blue is for hexes to arrive at shore level or for hexes to be essential in blocking line of sight from hexes that are already blue.

Eventually the entire puzzle will be blue.

It works a bit like the notion that a sculpture is already there within a block of marble, you just have to chop away the right parts to release it.
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It's a pretty reliable method, though perhaps overcomplicated for easier puzzles. It helped me solve every last one on this site.

That is an interesting method. I guess it may be similar to what I sometimes use (I also often work top down), but not in such a structured way. I like the sculpture analogue...

I'm wondering... what do you do when none of the remaining numbers see too much? (But some of them see too little, otherwise the puzzle would be solved, obviously.) Or does that never happen?

Whoops, I can't believe I forgot to mention that part, because it does happen all the time, numbers not seeing enough. When that happens, there are some situations that occur regularly and have obvious ways to deal with them, and some less obvious ones that require a bit of logical creativity. The more hexes you've already managed to paint blue, the less options you have, and the easier it becomes obviously.

For example, for hexes close to one or more edges of the puzzle you can often figure out you need to clear a path of at least a certain number of hexes into one direction because all other directions combined can never add up to more than a certain number of visible hexes. The cascading effect caused by the maximum height difference of 1 often produces a lot of new clues, including hexes that suddenly see too much because something in their line of sight got reduced in level.

There are really too many possible situations and ways to deal with them to get into unless I'd want to produce a complete guide. The mechanism of only ever having to reduce heights gives many of these puzzles a nicely flowing solving process.

One last point: there is one puzzle (I can't remember which one) that defeats practically all of the typical situations my method runs into. That one had me thinking the way I was doing it had been anticipated.

I see. I'm glad to hear there's still logical creativity involved, and it's not just a completely mechanical process.

I think I'll try it out and see how it feels in practice.

Forseti wrote:

That one had me thinking the way I was doing it had been anticipated.

Nope. Just pure chance.

Maarten wrote:

Nope. Just pure chance.

^ Assuming it was made by me, that is...