# PSE Advent Calendar 2023 (Day 11): Divide the cookies and milk

This puzzle is part of the Puzzling Stack Exchange Advent Calendar 2023. The accepted answer to this question will be awarded a bounty worth 50 reputation.

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Santa’s little helpers come in all shapes and sizes. Their upkeep requires a lot of cookies and milk, both as reward for their tireless efforts and to provide energy allowing them to perform their tasks. Divide the table below into regions along grid lines so that every region contains one-half milk glasses (shown as white squares) and one-half cookies (shaded squares). The two halves must be the same size and shape, although they can be rotated and/or mirrored compared to one another. Some glasses/cookies are allocated to a helper of a certain size; these are marked with numbers in the grid. For example, a number 3 in a cell would indicate that the region that cell belongs to contains 3 cookies and 3 glasses of milk. Regions can contain any number of numbered cells, including none. Can you divide the cookies and milk so that everyone is happy?

TL;DR Nikoli’s Double Choco.

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Solution:

Path:

First, notice that there's only one way to divide up the milk region on the side into a 5 and a 3, and there's only one way to attach cookie regions to each of them; if the size 5 cookie region were shifted down one, there would be no way to fit in a straight triomino of cookie below it:

That has now split the cookies on the right into two distinct regions. The one below it has size 5 and is also adjacent to a 5, so we do have to use the full thing, and there's only one way to fit that shape into the milk. We can also look at the 4 above it: the only way to extend it is upward until it reaches the other 4, and the only way to fit a straight tetromino into the milk region is to have it be vertical as well:

Let's finish off that right-hand cookie cluster. The 3 will either divide the 4s into undersized regions or not reach the milk unless it goes straight up, at which point there's only one way to include the milk. Now the other two 4s over there have to be part of the same region, and it has to be a J rather than a Z as a Z would trap the resulting single cookie with no access to milk. That J tetromino has to reach the 4 in the milk, as not doing so wouldn't leave enough room for another 4-region plus a 7-region in this milk:

That single cookie now only has one way to connect to milk, and what do you know, that leaves exactly 7 milk cells! Now how do we connect that to cookies? There's a 7 hanging out in there, and while there is technically enough room to fit two rooms of size 7 while still fitting the 4 and the 1 that are also required, it's not a shape that can fit into the milk on the left. Therefore, it must connect to the milk only on the lower left so that it can reach that other 7, although we can't quite determine its orientation yet. This does mean, though, that the single milk cell has only one way to go, and the cookie in the bottom right of the tens digit has to escape through the 4 along the bottom of the grid, then connect to a straight-line milk region:

Now it gets a little tricky. There are two ways to resolve all the 5s in the top left. One of them is a neat little donut shape that surrounds a munchkin in the middle, as you can see marked in red below, but that leaves only one way to resolve the 2, which in turn, makes it impossible for the milk 4 and the cookie 4 to be the same shape:

Thus, the 5s have to connect without donutting, and in order to avoid the same situation with the 4 below, the 2 has to connect upward rather than downward:

Now, it's pretty clear that the 4s have to be squares, and that, finally, disambiguates the 7 at the bottom, completing our puzzle!