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# Puzzle 40: Singularity - The Zotmeister

## Aug. 6th, 2006

### 02:31 pm - Puzzle 40: Singularity

Among the Nikoli dynasty puzzles, Hitori reigns supreme with a brutal iron fist. Its logic is harsh; its observational requirements are far uglier. It took me quite a long time to figure out how to even try to make one of these, especially given the unwritten rule that there cannot be more cell values than the side length of the grid. It's very tempting to overhaul the design for the sanity of myself and my solvers - lift that restriction, only put values in cells that need them for deductive purposes, that sort of thing - but as this is part of the Puzzle Japan series, such tweaks shall have to wait. I did go ahead and use letters instead of numbers so that I wouldn't need multiple characters in a cell, but it doesn't change the puzzle in any functional manner (just like To Each Their Own, you don't even need to know how to read them). At any rate, my Singularity has the same rules as Nikoli's Hitori, and these two have the same format. If you wish to run away, now may be a suitable time.

Still here? Okay, but, to quote Garth Marenghi, "don't say I didn't specifically warn you, because I just have and that means you'd be lying":

The grid on the left is a Singularity puzzle in its unsolved state. The grid on the right is its solution.

The object is to eliminate cells from the grid in such a manner as to leave each row and each column with no duplicated symbols within; no two orthogonally adjacent cells may both be eliminated, and all non-eliminated cells must be orthogonally contiguous.

The rules are simple. The logic isn't. But just in case you disagree with me about the rules part:

1) Every cell of the grid is either to be eliminated or left intact. To give them names, I will call eliminated cells "quarantines" and non-eliminated cells "open cells"; quarantines are to be shaded in and open cells are to have circles placed in them (around any symbol within). Mark every cell of the grid as one or the other following the remaining rules, and the puzzle is solved!
2) You cannot have two open cells that are both in the same row or column that have the same symbol in them. For example, the sample puzzle has two 'E's on the top row; at most one can be open. (So if you find a cell must be open, shade in any cells in the same row and column that have the same symbol, as they must be quarantines.)
3) You cannot have two quarantines that share a side. They can touch at corners, but can't be right next to each other. (So if you find a cell must be a quarantine, put a circle in the cells on each side of it, as they must be open.)
4) Every open cell of the grid must be part of a single polyomino - a group of quarantines connected through their corners can't be allowed to sever the grid (either by forming a loop or a chain that touches the outer border twice or more). For example, if the cell in the bottom-right corner of the sample solution were a quarantine, the cell above it would be (orthogonally) disconnected from the other open cells, which is illegal.

Why I didn't simply cut and paste rules from any of the previous dynasty puzzles (namely Smullyanic Dynasty, Echolocation, Room Reason, and even Ariadne's Lament if you think about it) is something I'm not entirely sure of. Anyway, I thoroughly expect most of you will need this if you haven't tackled a Hitori before:

(Start of sample puzzle solution)
Right, so we can't leave any duplicate letters in any row or column... but how do we know which to keep and which to ditch? Well, in order to get a foothold to start from, there are a few patterns one can search for. Here, the bottom-left corner provides the breakthrough: there are two 'A's along the bottom row next to each other, and each has a 'B' above it. This sort of "striped" area has to end up checkered, and given how it's parked in the corner of the grid, it only has one way of running. Let me explain it this way: pretend the 'A' in the bottom-left corner is open. Since the 'A' right next to it is in the same row and has the same letter, it would need to be a quarantine. This makes the 'B' above it open - since quarantines can't share sides - and lastly that makes the 'B' to its left a quarantine. The result is that the bottom-left corner cell is open, but the cells above it and to its right are both quaratines, separating it from the rest of the grid. This is illegal. Therefore, the bottom-left corner is a quarantine, as is the cell diagonally up and right from it; the other 'A' and other 'B' those are next to must both be open.

Still with me? You're a brave one. Okay, there are four more cells that can promptly be marked as open. Two are next to quarantines, so those are obvious. The other two are the center cell of the leftmost column and the center cell of the bottom row. If either of those were quarantines, they'd illegally isolate an open cell. As it happens, the former leads to another quarantine: there's another 'C' at the far right of the center row. That sort of thing you just plain need to watch out for. On hard Hitori puzzles, I figure I typically spend at least quintuple the time searching for this sort of "transference" than I do actually solving anything logical. Good luck.

...You haven't run away screaming? Very well. I did not make this sample puzzle easy, because if anyone's to stand a chance at all of getting through one of these with their sanity intact, they need to learn and learn well how to recognize and utilize "contingencies" in the grid. The very next step requires one. Look at the last quarantine I placed. Diagonally up from there is an 'E'; diagonally down from there is an 'E'. These are both in the same column, so at least one of them is a quarantine. We may not know which is a quarantine yet, but contingent upon either being shaded in is that the grid's center cell must be open! See how we'd have a chain of quarantines going all the way across the grid if it were a quarantine instead? It'd start at the 'A' in the bottom left, move up-and-right to the 'B', up-and-right again to the center cell, then to whichever of those 'E's is a quarantine, then finally to the 'C' on the center right. This chain severs the grid; open cells above it cannot be connected to open cells below it. This is very much illegal. Marking the center cell of the grid as open, the entire remainder of the puzzle can be solved using the same logic of the first two paragraphs of this explanation; incidentally, learning which of those two 'E's is a quarantine is one of the last things uncovered.

If you could get through all that, you may just be able to solve these. You'll likely need to teach yourself a few other patterns and tactics, but at least you should now know some of the basics.
(End of sample puzzle solution)

I don't consider myself very good at making these; I strongly believe there's some part of "the memo" I didn't get, some tactic that eludes me. This was a pain to build - not in generating the logic chain, but in enforcing it while still putting a letter in every cell and restricting myself to 'A' through 'L'. I may never put myself through that again, and just make any future puzzles of this design my way, but time will tell. - ZM