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# Puzzle 20: Archipelago - The Zotmeister

## Nov. 10th, 2005

### 03:09 pm - Puzzle 20: Archipelago

Once upon a time there was a clever little puzzle named Islands in the Stream. And it was good. There was also this neat little puzzle named Block Party. And it was fun. Then one day, apparently some evil scientist wanted to see what their offspring would look like, and created Archipelago: the bastard cousin of island puzzles. Although many concepts are familiar, the rules are just different enough to really screw with the head of someone who's been solving other puzzles.

The grid on the left looks like it could be an Islands in the Stream puzzle, but don't be fooled: it's an Archipelago puzzle. The grid on the right is the unique solution. (If you were to try to solve it as an Islands in the Stream puzzle anyway, you may find it would have two solutions. Only two cells remain ambiguous, but that's all it takes to invalidate it.)

Each cell of the grid is either "land" or "water"; the objective is to determine which for all cells. Every land cell belongs to exactly one "island": a rectangle of land cells that contains at most one numbered square and, if it has a number, as many squares total as that number. Islands may never be orthogonally adjacent, but ALL islands MUST be diagonally contiguous. The water cells must contain no numbers and comprise no solid 2x2 squares ("pools").

"...What the?! Now e's saying 'diagonally contiguous'? What has gotten INto Adam?!"

Yeah, well, like I said, it screws with people's heads. Here, I'll try to sort it out for you:

1) Each cell of the grid is either a "land" cell or a "water" cell. Fill them all in to solve the puzzle. (Rather than using colored pencils, I recommend simply dotting the unnumbered land squares and shading the water squares as you go along.)
2) Each numbered cell is land. (No need to put dots in them.)
3) For each number, there is one "island" of land cells. In this puzzle, islands are strictly rectangular, so for each number you need to portion off a box whose area is that number. Islands may not have multiple numbers.
4) Each land cell is a part of one and only one island; islands cannot overlap, nor can they touch each other EXCEPT at their very corners.
5) ALL islands must be connected into a single "archipelago", such that all are diagonally contiguous. In other words, not only CAN the islands touch at corners, but they HAVE to, such that one could travel from any arbitrary island to any other staying on land by solely passing through island corners.
6) No "pools" are allowed: you can't have four water cells making a two-by-two square anywhere in the puzzle. Even if it's part of a larger rectangle of water cells, it's still a pool and it's still illegal.

Note that there's no restriction on having an island WITHOUT a number. That's part of the puzzle: you may find that in order to prevent pools and to keep the archipelago connected, you may find the need for an island (of deducible area) to be in a numberless spot. Don't forget that ALL ISLANDS MUST BE RECTANGULAR, even unnumbered ones - that's how you can deduce the unique solution!

(Start of sample puzzle solution)
Certain Islands in the Stream and Block Party tactics still work here. The easiest place to start is the '1' in the bottom-right corner; that's a complete island, and as such needs a water cell directly above it and to its right. The '3' in the upper-left is not much more difficult: since its island needs to be a rectangle, and that rectangle would lie right next to the '4' if it ran to the right, it must instead run down, with three water cells placed to its right. Now both of these islands need to be connected into the archipelago by their corners; each only has one free corner to step out from, and it just so happens that they lead to the same cell. Marking that as land, the two leftmost columns of the grid are filled already!

Now that land cell is sitting right next to a '2', defining that island. Marking the four water cells that share sides with that island, we get an elbow: the center cell of the second row from the top must be land - we'd have a pool otherwise. This defines the island the '4' must belong to; it obviously needs to be a square, but if it were a square in the upper-right corner, that land cell we just added would be illegally sitting next to it. Instead, the island needs to include that cell. Three more water cells around the island can be marked, and noting that the island has only one free corner left, the center cell of the rightmost column must be land.

All that remains is to legally connect the two portions of the archipelago, and there's only one way to do it. Diagonally down and right from the '2' island is the only liberty the network of '1', '2', and '3' has; that cell must be land. Compare it to the land cell diagonally out from the '4': they can't be part of the same island because of the water cell in the way of putting them in the same rectangle. Now if the lower land cell were to extend to the right, it would lose its only liberty and split the archipelago; therefore, the bottom-right-corner cell is water, and the one above it land.
(End of sample puzzle solution)

I was surprised how easy it was to make these, especially the sample puzzle - making a useful Block Party sample, in comparison, was far trickier. Hopefully this one will get you thinking. My email address is on my User Info page if you'd like to share your solution with me. I have the occasional habit of sharing something in return. Any generic comments about the design itself can be posted right here. But hey, don't blame me - it's not my original design. I figure someone at Nikoli was having a VERY BAD DAY when e came up with this one. - ZM

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