Double Chocolate: Happy Valentine's Day!

Question

Hikouki (@flight_float) over on Twitter issued a challenge to make a Double Chocolate logic puzzle with a fixed grid pattern, and lots of logic puzzle makers responded to that challenge! Here's my take on it.

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Rules of Double Chocolate:

• Divide the grid into regions by drawing along some of the dotted lines.
• Each region should have one group of gray cells and one group of white cells; both of these groups should be connected.
• The two groups in a region should be the same size and shape (but they may be rotated or reflected).
• Each number tells the size of each of the groups in its region.

^{A convenient interface for solving this puzzle is available here.}

Answer 1

Here's the solution:

And here's the explanation:

(Note: In this explanation, whenever I use the phrase "$ n $ region," I mean a region with $ n $ gray and white squares that satisfy the constraints of the puzzle.)

We start off with some easy deductions: lines must separate regions with different numbers, the 1 in the upper right is forced, and the 2 in the center left must be constructed upward:



Now, assume that the 2 in the upper left includes the white square immediately above it. This forces the 1 and 4 regions to be constructed as shown. Then, there is no way to construct the two 2 regions without violating the "each region has an equal number of white and gray squares" rule.



Thus, the 2 region must include the white square immediately below it. This forces the 4 to include the white square directly above the 2, which in turn force the constructions of the 1 and both 2 regions. This leaves two options to construct the 4 region, but only one of them is valid (the other leaves one gray square by itself, which is not allowed):



Now, we focus on the 14 region. (Credit goes to Avi in chat for helping me with this part of the explanation.) Intuitively, it must be connected to the block of 14 white squares in the bottom left. To make this rigorous, we will show that we cannot have a region of 14 white squares anywhere else. If we try to make a 14 white square region in the top/upper right, then we block the 4 in the upper right from reaching white squares. If instead we try to free up that 4 by making a 14 white square region in the bottom right, then we block the 4 in that part of the grid from accessing white squares:



Thus, the 14 region must connect to the 14 white square block in the lower left. This also forces the shape of the 14 gray square block, as well as the construction of two 1 regions:



Now, we focus on the 4 region in the lower right. Clearly, it must use white squares in the bottom right, but it cannot divide that area of the grid into two, otherwise there will be isolated regions of white squares. Thus, the only way to construct the 4 is to take the white squares in an L-shape, just straddling the edge of the grid. This will also force the construction of the 2 region in the bottom right:



This leaves us with a long L-shaped white square block in the bottom right that is 3 squares wide and at least 5 squares tall. There is only one way to fit such a shape into the black squares and have it connect to the white squares, as shown:



Note that the above construction also forces the construction of the remainder of the 4 region. At this point, we are pretty much home free, as the remainder of the deductions are fairly intuitive. First, complete the upper right portion:



Then, note that there are two possible 1 regions that can be created from the lone gray square. Only one of these allows us to construct the remaining regions so that they satisfy the rules, and this is our final answer: