The fundamental rule is that board generation has one, and only one solution. It doesn't sound that insightful on the surface, but it is that assumption which led me to discover important advanced strategies, as well as spot other smaller, esoteric cases. This post covers one of those esoteric cases, but it illustrates the rule well.
Let me explain with an example, of a partially completed board:
🔵5️⃣4️⃣4️⃣🔵
⚫️🔵🔴🔴⚫️
⚫️🔴⚫️⚫️4️⃣
2️⃣2️⃣⚫️⚫️3️⃣
🔴⚫️⚫️5️⃣⚫️
Specifically, let's focus on the leftmost 2️⃣, and ignore the irrelevant parts:
🔵5️⃣🔵🔵🔵
⚫️🔵🔴⚫️⚫️
⚫️🔴⚫️⚫️⚫️
2️⃣⚫️⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
I am reasoning that it should be filled in like so:
🔵5️⃣🔵🔵🔵
🔴🔵🔴⚫️⚫️
🔵🔴⚫️⚫️⚫️
2️⃣🔵🔴⚫️⚫️
🔴⚫️⚫️⚫️⚫️
How can I be so sure? Suppose immediately above the 2️⃣ lives a red wall. The dot above that could be either red or blue (denoted by ❔)
🔵5️⃣🔵🔵🔵
❔🔵🔴⚫️⚫️
🔴🔴⚫️⚫️⚫️
2️⃣⚫️⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
In other words, the board would have two solutions, which breaks the rule that boards have a single solution.
So it must be a blue dot above the 2️⃣.
🔵5️⃣🔵🔵🔵
⚫️🔵🔴⚫️⚫️
🔵🔴⚫️⚫️⚫️
2️⃣⚫️⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
From here, conventional rules can be applied. Looking further north would exceed the 2️⃣, so it must be a red wall.
🔵5️⃣🔵🔵🔵
🔴🔵🔴⚫️⚫️
🔵🔴⚫️⚫️⚫️
2️⃣⚫️⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
And the 2️⃣ has only one direction left to look in.
🔵5️⃣🔵🔵🔵
🔴🔵🔴⚫️⚫️
🔵🔴⚫️⚫️⚫️
2️⃣🔵🔴⚫️⚫️
🔴⚫️⚫️⚫️⚫️
Let's add back the other dots and solve it, to see if it works!
🔵5️⃣4️⃣4️⃣🔵
🔴🔵🔴🔴⚫️
🔵🔴⚫️⚫️4️⃣
2️⃣2️⃣🔴⚫️3️⃣
🔴⚫️⚫️5️⃣⚫️
(Filling in the 2️⃣ and 5️⃣)
🔵5️⃣4️⃣4️⃣🔵
🔴🔵🔴🔴⚫️
🔵🔴⚫️🔵4️⃣
2️⃣2️⃣🔴🔵3️⃣
🔴🔵🔵5️⃣🔵
(3️⃣ can see all dots)
🔵5️⃣4️⃣4️⃣🔵
🔴🔵🔴🔴🔴
🔵🔴⚫️🔵4️⃣
2️⃣2️⃣🔴🔵3️⃣
🔴🔵🔵5️⃣🔵
(Only one direction left for 4️⃣ to look in)
🔵5️⃣4️⃣4️⃣🔵
🔴🔵🔴🔴🔴
🔵🔴🔵🔵4️⃣
2️⃣2️⃣🔴🔵3️⃣
🔴🔵🔵5️⃣🔵
Yay! It seems like this idea might help complete more boards.