June 27th, 2005


Puzzle 12: Echolocation

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:

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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