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Utility Graph

You cannot perfect this map (with 99.4% prob).

This is the classic "utility graph", which is one of the simplest graph that is not a planar graph, i.e., you cannot connect every utility provider to every house without making a cross.

In this setting, each power plant island represent a utility provider (water, gas, electricity). And each portal islet represent a house in the original math's puzzle.

However, this game doesn't allow one house to be connected by multiple sources, so I've expand "a house (in the graph theory puzzle)" into a village of 18 villae. (Floating islands of the same color = a village.)

Since the game just (uniformly?) random a color for a villa, one village will likely require all 3 colors at a probability of

S(18,3) * 3! / 3^18 = 99.8%

where S(n,k) is a Stirling number of the 2nd kind. (Read more about this in Coupon collector's problem.)

There are 3 villages, so it is (99.8%)^3 = 99.4% probability of all 3 villages require all 3 colors. Hence, improbable to perfect this level.