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Puzzle 21: Room Reason - The Zotmeister

solving the puzzle of life one entry at a time

Nov. 30th, 2005

05:59 pm - Puzzle 21: Room Reason

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This title is the first I've presented here that I didn't come up with myself; rather, it is how Babelfish translates Nikoli's title, Heyawake. I've been told that the translation is inaccurate - "divided rooms" would be correct - but given the alliteration and the apparently unintended synonym of logic appearing within, I decided this translation is better than the "right" one, so I'm keeping it.

This puzzle has been getting a lot of attention lately... in Japan, at any rate. Proving to be one of Nikoli's most addictive creations, it was recently the subject of a sizeable book all its own, and is being advertised as an upcoming offering at Puzzle Japan. Just last night I finished my copy of Heyawake 1, meaning I have over a hundred of these solved. I learned many things going through that book, but mainly that I still have much to learn. Thankfully, I have four other all-Heyawake books to continue learning with. But first, here's one for you.

Yes, it's another dynasty puzzle. GLmathgrant pointed out to me why this mechanic is so popular in Japan: it's the set of rules used to construct Japanese crossword grids!

On the left is an unsolved Room Reason puzzle; on the right is its unique solution, rendered in turquoise because I haven't used it yet and I like how it's spelled.

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. The grid is divided into "rooms" by the thicker borders: numbered rooms must have exactly that many pillars within; nowhere in the grid may there be an orthogonal line of consecutive open cells that starts in one room, crosses straight through a second, and ends in a third (a "spanner").

Hopefully your skills in reading instructions have improved lately, but I'm still with you if you have trouble parsing my one-paragraph definition:

1) Think of the grid as a house. Most of it is open space, but some cells are "pillars" - they contain support posts. 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. You're probably sick of my using that phrase by now, but it remains the simplest way of detailing the concept. Working on Wikipedia articles, however, I came up with what may be the second-simplest way of getting it across: all open cells must form a single polyomino.
4) See all those extra-thick borders? Yeah, three times thicker? Those are walls, and they're portioning the house into - say it with me - "rooms". Now some rooms have numbers in them; those rooms have that many pillars in them. No more, no less. Numbers are always written in the upper-left-corner cells of their rooms, for no reason other than consistency. Don't let that throw you; the numbers are always referring to their rooms. Numbered cells themselves can be open or pillars just like any others.
5) No "spanners" are allowed. A spanner is any place where a line of open cells runs up, left, down, or right from one room straight through another room completely, and into the room beyond. The (heavily abstracted) idea is that the roof would be unstable in the room the open cells span across. For example: look at the leftmost column of the sample puzzle grid. If the top four cells of that column were all open, that would be a spanner - it would start in the upper-left-corner room, run straight through the room below it, and end in the room in the bottom-left corner. That's illegal. The outer border of the house itself is perfectly stable; note that in the sample solution, the entire rightmost column is all open, but that's fine since there's only two rooms on that side, and by definition a spanner needs three.

This particular sample puzzle is fairly simple, but it does have a lesson to teach, so if you're having trouble figuring it out yourself:

(Start of sample puzzle solution)
One of the big keys to solving a Room Reason puzzle is understanding how pillars are forced into a checkerboard pattern in small areas. If you remember Smullyanic Dynasty, some of the rules there have equivalents here! In fact, it's just such an equivalent that I'm going to use to start this puzzle with. The bottom-left-corner room of the sample puzzle is a 2×2 room with a '2' in it. Now since pillars can't share sides, the two pillars in this room need to be in opposite corners of it. But if the two pillars were in the top-left and bottom-right corners, they'd be trapping the cell in the bottom-left corner of the grid! Since all open cells must be connected through their sides, the two pillars will need to be placed in the other two corners. Again, since pillars can't share sides, there are four dots that can promptly be added as well.

Now I examine the bottom row. The center cell of the bottom row can't be a pillar - it would be trapping the cell to its left - so it must be open. (Remember - every cell is one or the other!) Now this sets up a "spanner threat": we have a room - specifically the center column of the grid - that has an open cell in it and an open cell alongside that one just outside. Since this room is only one cell wide, that's two parts to a spanner right there. If the cell that's second from the right on the bottom row were to be open, that would complete the spanner, and that's a no-no. That cell must be a pillar instead. Surrounding it with dots, I have nine of the twenty-five cells of the grid marked.

Next I'll visit the leftmost column. As perhaps you've already marked, the center cell of this column must be open to prevent trapping the cell under it. Now look at the number in this room and the room above it. They both have '1's, and they each have two cells left for their pillar to be in. But see how the fact that the rooms are alongside each other forces their pillars to be diagonally opposite each other? You may have guessed it - this makes them function just like the very first room I solved! As this "makeshift 2" is also in a corner of the grid, it needs to be solved in much the same way. Filling in all the appropriate dots (including the center cell of the top row), I find another spanner threat along the top row, just as I found on the bottom row, and I deal with it in the same manner.

Now consider the center cell. Yes, of the whole grid. Can this be a pillar? No, and the reason why is that it severs the dynasty. ...Ahem. I mean it breaks the house in two. There would be a diagonal line of pillars running from the top-left corner to the bottom-left corner. The open cells on each side of that line could not connect with each other. Marking the center cell as open, I find one last spanner threat. That places the last pillar; of the three remaining unmarked cells, one shares a side with that pillar and the other two (in the rightmost column) would trap corner cells if pillars.
(End of sample puzzle solution)

Although not terribly strenuous by typical standards, this particular puzzle may prove challenging to the inexperienced, and has at least one fairly tricky moment built into it. I'll just give you the trial by fire, as it were, and let you try to learn the reasoning behind these rooms yourself. Email me if you manage to solve it. In fact, email me if you manage to get stuck, and I might even help you. Maybe. If I have the time. At any rate, expect quite a few of these in the future, since I've got them on the brain. Or post a comment here begging me not to, or something. If you don't like this design, I have an old standby coming up, like, tomorrow. - ZM

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