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Infinity Loop

A problem inspired by the game, Infinity Loop, for iPhone and Android.

Consider a subset $$S$$ of an infinite lattice grid, consisting of all points $$(x,y)$$ where $$x,y$$ are integers ranging from $$1$$ to $$10$$ inclusive.

Let each lattice point be assigned an integer $$a_{x,y}$$ ranging from $$0$$ to $$4$$ inclusive. Define a connection of $$S$$ as an operation in which you add line segments between two horizontally or vertically adjacent lattice points, such that the point $$x,y$$ is connected to exactly $$a_{x,y}$$ segments for all points in the set. Each pair of adjacent points may have at most one line segment connecting them. Points in the corners can have at most 2 line segments, points on the sides 3, and points in the middle 4.

Given that you know all the $$a_{x,y,}$$, is it always possible to tell whether a connection of $$S$$ exists?

Note by Daniel Tan
5 months, 1 week ago