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Puzzle 12: Echolocation - The Zotmeister

solving the puzzle of life one entry at a time

Jun. 27th, 2005

04:09 pm - Puzzle 12: Echolocation

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After learning the rules to a puzzle Nikoli calls Where is Black Cells, it took me all of fifteen seconds - tops - to come up with a better name. If some of the rules of this puzzle sound familiar, that's because they are: this puzzle uses the Dynasty mechanic I borrowed for Smullyanic Dynasty.

Yes, those numbers are too big for an Islands in the Stream. The left side is an unsolved Echolocation puzzle; the right side is the same one having been solved and marooned.

Every cell of the grid is either a "pillar" or "open"; the objective is to determine which for all cells. Pillars may not be orthogonally adjacent; all open cells must be orthogonally contiguous. Cells containing numbers are open; each number is the quantity of open cells orthogonally radiant from it, including its own cell.

...What do you mean you don't know what "orthogonally radiant" is? Just because I made it up doesn't mean it isn't perfectly clear♥ ...I'll give you the classic list:

1) Think of the grid as a cave. Most of it is open space, but some cells are "pillars" - solid rock. All cells are one or the other. The object is to find all these pillars (and, consequently, where all the open space is). I suggest shading in pillars as you find them, and (just as importantly) marking cells you know must be open with a dot - this is what I've done with the sample puzzle.
2) Pillars never share a side, but they can touch at corners. (So if you place a pillar, you can mark dots in the cells that share sides with it.)
3) All open cells - that is, all that aren't pillars - must be orthogonally contiguous. There's that phrase again. I defined it back in Puzzle 1, but here's a way to define it I haven't used yet: if you were a chess rook sitting on one of the open cells, and had unlimited turns, you would have to be able to ultimately reach all other open cells without moving onto or crossing any pillars (as if they all had friendly pieces sitting on them that couldn't move). If you don't know the rules of chess, you should learn them, since you'll need them for an upcoming puzzle.
4) Every cell that has a number in it is open - it's not a pillar. (Don't bother dotting it - it's rather obvious.)
5) Think of the cells with numbers as having bats in them at the moment. (Animals, not sports equipment.) Bats can tell how far away walls are via echolocation, and they're passing the knowledge on to you. Each number is how many open cells exist total between the one it's in and either the outer border or the first pillar it encounters in each of four directions (up, down, left, right). This total includes the cell the number is in, but not the cells pillars are in - the numbers are stricly open-space-only counts. To continue the chess rook analogy, a rook sitting on a number would be able to attack one less than that number of cells if the aligned pillars were occupied with friendlies and it had no other obstructions.

Since that last rule is easier demonstrated than said: consider the '5' in the sample puzzle solution. Left from the '5', we immediately hit the outer border of the puzzle - zero there. Up, we immediately hit a pillar - zero there as well. Down, there are two open cells, and then we hit a pillar - two in that direction. Right, there are two open cells before a pillar - just those two count; the one behind the pillar doesn't. Add in one for the numbered cell itself, and we get 0 + 0 + 2 + 2 + 1 = 5, which matches the '5' in the grid. All the numbers have to add up like that. For the '8', we get 2 + 2 + 2 + 1 + 1 = 8. I'll let you check out the other three givens if you are so inclined.

If that still stumps you, then read through this and maybe you'll get the idea:

(Start of sample puzzle solution)
I start with the two '2's on the right side. If that cell between them were open, then they'd both be able to "see" three cells. That's too many, so the cell between them is a pillar. Remembering pillars can't share a side, the cell to the left of that pillar is open and gets a dot.

Now look at the '8' in the center. There cannot be any more pillars aligned with it - it needs all eight cells it could still have! The entire center column is therefore open, as is the center row apart from that already-placed pillar on the right. That's six more dots to add.

The '3' on the left side is now "defined", as I like to say - with the cell above it marked as open, and the cell above that containing a number, that's the three open cells it needs to see. It can't be allowed to see any more, so we can add three pillars: just below the '3', just to the right of the '3', and just above the '5'. There are also two dots we can add next to those pillars.

We can now define the '5'. It currently sees three cells, but has only one direction it can get more in - right. Dotting the cell to the right of the '5', it gets the cell beyond it as well, making the five it needs. We add a pillar to the cell beyond that one (to the left of the upper '2'), and then a dot above that pillar.

We now go back to the '2's to finish. The upper '2' has only one direction left it can look in - up - so the cell above it is open and gets a dot. As for the bottom '2', it can't be allowed to look left, as it would see three cells, so the cell left of that '2' is a pillar. The last two cells of the bottom row must both be open: the leftward one because it's under a pillar, the rightward one (the bottom-right corner) because it would sever the Dynasty - um, I mean it would break the cave in two. Well, that, and the lower '2' needs to be able to see it. The puzzle's solved at any rate.
(End of sample puzzle solution)

As is often the case with my puzzles, larger doesn't necessarily mean harder. Don't let the size of this first one intimidate you; it's fairly straightforward. Email me your solution if you solve it; feel free to make comments about it here as well. - ZM

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