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Puzzle 41: Totally Rooked - The Zotmeister

solving the puzzle of life one entry at a time

Aug. 7th, 2006

10:36 am - Puzzle 41: Totally Rooked

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When glmathgrant beat me to a clever retitle for Nikoli's Light Up - namely Process of Illumination - I felt defeated. E offered to let me steal it from em, but I didn't want to do that. It was just recently that I came up with not only a name I liked, but in fact an entire reskinning of the design. It quickly occurred to me that I didn't need to stop there; here I present an original generalization of Light Up, complete with my signature work-around-the-exception mechanic.

The grid on the left - and the text beneath it - are an unsolved Totally Rooked puzzle; the grid on the right shows the solution to same, rendered in gold. (Technically, the light yellow crosses are not part of the solution, but you know me, I hate to leave anything completely blank in puzzles like this - I like to know when I've finished.)

Every cell of the grid is either a "wall" (blackened in, possibly with a numeral) or "open" (left blank). Surprise - the object is not to determine which is which♥ In fact, that knowledge is all given. The object is to place the chess pieces listed below the grid, each in their listed quantities (some of which may be unknown), such that every open cell is either occupied by exactly one piece or is attacked by at least one piece. No piece may attack any other. Walls block the attack ranges of the pieces as if they could not be moved through (which of course doesn't affect knights); multiple pieces in cells orthogonally adjacent to a single wall must all be the same type. If a wall has a numeral, exactly that number of pieces must be in open cells orthogonally adjacent to it.

Sometimes the simple puzzles have complex-to-phrase rules. Let me see what I can do in terms of clarification:

1) All the open (white) cells of the grid need to be "rooked" in order to solve the puzzle. To rook a cell, one needs to either put a chess piece in it, or have it be attacked by a chess piece in some other cell. Think of the pieces as painting the cell they are in and every cell they attack - paint all the open cells without violating the following rules, and the puzzle is solved! I suggest crossing out rooked cells as you determine them, except of course for those that hold a chess piece (which makes the crossout redundant).
2) Walls (the black cells) block the (theoretical) movement of the pieces, just like a friendly piece does in chess. For example, the sample puzzle solution has a rook in the upper-right corner; it only rooks two cells, its own and the one to its left; it can't attack beyond the walls to any further left or down. Naturally, just like in chess, a wall between a knight and its destination doesn't block the knight.
3) Only the chess pieces listed below the grid must be used for this task, and their usage must match the exact quantities given for each. Often, some quantity will not be known in advance, but don't worry - to obey all the rest of the rules, the quantity will be forced, and the puzzle will still have a unique solution. For example, the sample puzzle must have exactly one queen and exactly one knight, and no other pieces but rooks. One cannot use pawns, bishops, kings, amazons, grasshoppers, marshalls, cardinals, dragons, cannons, or any other kind of piece. As for the rooks, I haven't given how many will be needed - could be a dozen, could be zero - but as the solution shows, four works, and its the only value that does.
4) Open cells can't hold more than one piece. Chess only allows for one piece per square, and so does this puzzle. Walls can't hold pieces at all.
5) Some walls have numbers in them; exactly that many open cells that share a side with that wall must have pieces in them. For example, the sample puzzle has a '2' on the top row, so exactly two of the three cells next to that wall must have pieces in them.
6) This is important: pieces cannot be allowed to attack any others. As you might imagine, close to the whole sample puzzle could be flooded with rooks otherwise. This means that once a cell is rooked, a(nother) piece can't be placed on it. (So if the rooked cells are crossed out - as I suggested after Rule 1 - they help show where pieces can't be placed.)
7) This is also important: If there are any pieces in cells that all share a side with a single wall, they must all be the same kind of piece. For example, that '2' on the top row must have two identical pieces next to it; since there's only a single queen and single knight for alternatives, those two pieces must both be rooks. Note that this rule applies whether there's a number on the adjacent wall or not. For example, the sample puzzle solution shows a knight next to the wall in the leftmost column and a queen next to the wall in the bottom row; neither of those walls could also have a rook next door.

It's really not as bad as it seems. As a matter of fact, the offering below only involves one type of piece - the rook - so Rule 3 isn't terribly useful for it and Rule 7 can be ignored completely for it. By all means, if you have a question about my rules, feel free to add a comment and ask. That goes for any of my puzzles. I'd rather help an interested person learn than scare them away.

Yes, I know "rooked" doesn't seem to apply when, say, a knight is used, but 'knighted' is definitely not the word for it. :) I use "rooked" consistently both to maintain the pun and also as a tip of the proverbial hat to Nikoli's original Light Up design, which is equivalent to Totally Rooked with only rooks (of unknown quantity). The implementation on Puzzle Japan was beautiful: click on an open cell, and not only would a light bulb (rook) be placed there, but the whole row and column or parts thereof it illuminated (rooked) would automatically be shaded green. This made the puzzle not only decidedly non-tedious, allowing concentration on logic, but also the fastest and most entertaining to solve on the site. It was definitely the most popular: I figure the average Light Up had twice the solvers of the average anything-else.

Helpful hint: I suggest putting a dot into open cells deduced to not hold a piece but aren't rooked yet - for example, any cell that starts next to a zero. Once the cell becomes rooked, place the crossout right over the dot. (The dot is redundant at that point, due to Rule 6.)

The sample puzzle probably isn't the best of introductions to this puzzle, but I have a lesson to teach with it:

(Start of sample puzzle solution)
As I noted in the rules, the '2' on the top row must have two rooks next to it, since there aren't two queens or two knights to satisfy that '2' (remembering they need to be the same piece since they're next to the same wall). The same applies to the other '2' on the right border. These '2's interact with each other, and I'll show you how: imagine there being a rook just to the left of the right '2'. That rook would attack two of the three cells next to the left '2'... and since pieces can't attack each other, this means only a single place for a piece would remain next to that '2'. That won't do. Dotting that cell, I see that only two cells remain next to the right '2'. Both get rooks. I then cross out every cell those rooks can attack: the cell to the left of the rook in the upper-right corner, the rest of the center row, and the rest of the rightmost column - seven crossouts in all. One of those crossouts is next to the left '2'; this leaves only two spaces next to it for its rooks. Marking what they cover, only three open cells remain unrooked.

I didn't need to guess that that cell couldn't hold a rook. I know a general rule that would allow me to dot that cell even if the other '2' weren't there! Try to find that rule, and try to figure out how far it can be generalized; it'll be useful for solving. In the meantime, I'll finish the puzzle: I need to put one queen into the grid, and of the three open cells remaining, there's only one it could be placed in without it attacking a rook. After placing it and crossing out the cell right above it, there's only one open cell left, and one knight left to be placed. I'm sure you can figure out the ending.
(End of sample puzzle solution)

My muse for this puzzle is hopefully immediately obvious. As this is part of the Puzzle Japan memorial collection, only rooks grace this grid, making the puzzle identical in function to Light Up as noted. Enjoy. - ZM