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This is a Trichain puzzle. The rules of Trichain are as follows:

  1. Numbers must be white, and match the size of their white islands.
  2. Each white island can only contain at most one number. Some white islands may have no numbers.
  3. Black squares must be part of L-trominoes.
  4. Black L-trominoes cannot be adjacent horizontally or vertically, but all of them must be connected diagonally. In other words, one standing on a black square must be able to visit any other black square via Chess King's moves, stepping on black squares only.

Here is an example Trichain puzzle, and its solution:

Here is the real Trichain. The only clues this time are 4s. Go for it!

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Overview

To be completely honest, solving this puzzle was mostly trial and error for me. I initially focused on the top left corner and throughout the entirety of the solution path I focused primarily on testing how the diagonal relationships impacted the surrounding areas.

The Top Row

I spent a fair amount of time here, initially focused on the top left corner with a setup involving:

Screenshot of the initial thoughts.

However, as time passed, and I became more and more frustrated with it, I had a realization that I was focused on one way and needed to try something new. I started focusing on ways to create the first 4 in the top corner, ultimately settling on the following as I placed and erased blocks along the top to simulate placing the L and what the impact below it was:

Screenshot of the solved top row.

Closing the Obvious

Certainly not obvious to everyone, but considering I had tried many solutions in the area by painting and erasing, I knew based on the top row that the diagonal connections would only support one configuration:

Screenshot of the closed middle left 4.

However, since I still had the right wall to solve, my suspicions of the lower left corner had to be validated by closing the right side.

The Right Side

Here, I started in the bottom right corner closing the right side with trial and error while taking into account the impact of the middle and left:

Screenshot of the solved right side.

Wrapping Up

Solving the right side confirmed my suspicions of the lower left corner due to the obvious connections required in the center:

Screenshot of the populated center.

The rest was as simple as following the diagonal connections:

Screenshot of the completed puzzle.

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    $\begingroup$ That will get you a (spoiler warning!) checkmark! $\endgroup$
    – Avi
    Sep 28 at 22:39

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