Consider an undirected graph $G$ where self-loops are not allowed. The vertex set of $G$ is $\{(i,j): 1 \le i \le 12, 1 \le j \le 12\}$. There is an edge between $(a,b)$ and $(c,d)$ if $|a - c| \le 1$ and $|b - d| \le 1$. The number of edges in this graph is _______.

The graph $G$ has $12 \times 12 = 144$ vertices arranged in a grid. An edge connects two distinct vertices $(a, b)$ and $(c, d)$ if they are adjacent horizontally, vertically, or diagonally (i.e., at Chebyshev distance 1).

We can find the degree of each type of vertex in the $12 \times 12$ grid:

1. Corner vertices (4 vertices): Located at $(1,1), (1,12), (12,1), (12,12)$. Each has $2 \times 2 - 1 = 3$ neighbors.

   $$\text{Sum of degrees} = 4 \times 3 = 12$$

2. Boundary non-corner vertices ($4 \times 10 = 40$ vertices): Each has $2 \times 3 - 1 = 5$ neighbors.

   $$\text{Sum of degrees} = 40 \times 5 = 200$$

3. Interior vertices ($10 \times 10 = 100$ vertices): Each has $3 \times 3 - 1 = 8$ neighbors.

   $$\text{Sum of degrees} = 100 \times 8 = 800$$

Using the Handshaking Lemma:

Thus, the number of edges is 506.