Solution to Nurikabe Puzzle #2

This is a step-by-step solution to JDA's Nurikabe Puzzle #2. It is a VERY difficult Nurikabe, made easier if you use "OWFU" inferences. (These are inferences that depend on the Puzzle being VALID: "Assuming X is white, then if Y works so does Z but the puzzle must have only a SINGLE solution to be valid; therefore X is NOT white."

In the solution to a nurikabe, all the cells end up painted either Black or White, but my nurikabe scripts allow you to set other colors for your own purpose while working on a solution. I've used those colors in these diagrams. White and Black are used for cells whose colors you've determined. Gray is for unknown cells. Light colors (Yellow or Green) are for cells I've recently proven to be White; a note may explain why. Light Gray cells are NOT proven to be White, but are colored for clarity: either that cell or one in its vicinity must be White, often to connect a Yellow cell to its group. Dark colors (Purple or Blue) are for cells recently proven to tbe Black. If Purple, it's typically provably Black because it's unreachable. Blue cells are usually known to be Black due to an OWFU deduction. In the first diagram, I've filled in some Blacks and Whites that are obvious. Then I show (with Yellow) cells that must be White to break up a possible 2x2 Black; these cells can be reached via only one group. Some of those deductions are best made in a certain order. Only the '5' marked 'A' can break up the possible 2x2 black far to its left; this means the '5' marked 'B' has to break up the 4x4 at the yellow near '5-A.' (The '4' to its right can't do it: it needs to break up a 2x2 below itself.) This leaves '5-C' as the only was to break up another (two) 2x2's. And this in turn leaves '4-D' to break up another 2x2. And finally that forces '5-E' to break up a 2x2 at the Green cell. A lot of progress due to these "Only One Way to Reach" 2x2's! But don't get your hopes up. :-) Things will get tougher!

I've filled in the Whites and Blacks just implied. Some unreachable (must be black) cells are shown in Purple. The cell colored Blue is reachable from the '3' but that wouldn't work: the '4' to its right could then grow only to three.

Now we make some OWFU deductions. Near the upper left there is a cell colored Blue which would connect two Black cells; do we need that connection or not? It's too early to easily deduce whether that connection is needed, but OWFU gives us our answer! Suppose that cell is White (i.e. that the connection is not needed). Then that 3-group could grow to either of two cells to its left which would work equally well. Either both lead to solution or neither does! Fine ... Except that we assume the puzzle constructor has done his job and there is EXACTLY one solution. Therefore the Blue cell cannot be White. (Actually there would be THREE possibilities if the connection weren't needed, since the connection, needed or not, wouldn't HURT.) For very similar reasons (though the details are more complicated), the Blue cell in lower left must also be Black. (Yet another inference of this type is available in the upper right, but we won't need it, so don't bother.) Finally a Green cell at the left must be White; otherwise no connection would be needed and the '3' could grow to the left in TWO satisfactory ways. We've made quite a bit of progress with three OWFU deductions, but we're about to apply an even more difficult one.

As you try to solve this puzzle, you will find that the '7' near the lower left is the hardest to deal with. There are many near-solutions where that '7' grows to only six instead of seven. And there are two black chains which can be connected along the bottom; Is that connection needed? Or might the 7-group sever that connection? OWFU to the rescue! The details may be tedious, but if the 7-group reaches the bottom row at all, it can do so in multiple ways. Therefore (OWFU) it does NOT reach the bottom row! There may be smarter ways to make some of these deductions (and OWFU can be thought of as cheating). But when I try to solve this puzzle I end up with cumbersome trial-and-error if I don't use these OWFU's. There's a light-blue cell marked 'X'. Once the Blues on the bottom line are in place, that 'X' cell can be shown to be Black by yet another OWFU: The 4-group can grow two different ways, and the two ways would work equally well if the 'X' were White.

We're essentially done. The smallish black group in the middle can connect only via the Blue cell. That expanded black group can connect only via two red cells; and a final connection uses three more red cells.

Is it beautiful? The black cells form a 3-coil spiral! A COMPLETE solution to a Nurikabe must PROVE that the discovered solution is unique. But we've not proved that at all; we've ASSUMED it for the OWFU inferences. The reader is free to pursue that proof; I'm happy now!

Is there a way to make progress without resorting to the OWFU's or a long trial-and-error study? Maybe, but I don't see it. Here's a diagram with cells painted Black or White when they can be deduced with relative ease, as we've seen above. But where do we go from here? If we could somehow deduce that the black chain in the bottom row connected to the black chain in the 1st column -- i.e. that the connections shown in Blue must be Black -- then the rest of the solution would fall into place quite easily. Otherwise, what do we have? Trial and error? If the green cell left of center is Black, we'll reach a contradiction (though not quickly), so we know that cell is White. That helps, but not too much. A much better idea -- and perhaps the most direct way to solution -- is to set the green cell below the center to Black. That leads quickly to a long series of deductions (including the forced connection shown in Blue), and reaches a contradiction: the 7-group at lower left attains size six only. So that green cell must be white; then everything else falls into place. (Proceed as above: the inner coil of the spiral has only one way to connect, then the enlarged group has only one way to connect, and so on.) I guess that's the best way. Just one trial-and-error excursion, and NO need for OWFU's. All of the "only-one-way-to-reach" deductions early on make me think this puzzle is elegant after all! Is it weird that my two favorites of all these Nurikabes I constructed are #1 and #2 ?!! The very first two I did. (The way the Blacks in the solution form three coils of a spiral was deliberate.)