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?

#Combinatorics

Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$