Expectation of Friends Count in Square Grid

To find the value of $7E(X)$, we analyze the grid of intersection points.

1. Understanding the Grid Structure

The given $6 \times 6$ square grid contains $7$ horizontal lines and $7$ vertical lines. The total number of intersection points (vertices) is: $\text{Total points} = 7 \times 7 = 49$

Two points $A_i$ and $A_j$ are "friends" if they are adjacent vertically or horizontally. Thus, the number of friends of a point corresponds to its degree in the grid graph.

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2. Counting Points by Degree (Number of Friends)

The $49$ points can be categorized based on their position in the grid:

1. Corner Points ($i = 2$ friends):

   • There are $4$ corners.
   • Each corner point is connected to $2$ adjacent points (one vertically, one horizontally).
   • Number of points with $2$ friends = $4$.

2. Boundary Non-Corner Points ($i = 3$ friends):

   • Each of the $4$ outer sides has $7 - 2 = 5$ non-corner points.
   • Total boundary non-corner points = $4 \times 5 = 20$.
   • Each such point is connected to $3$ adjacent points.
   • Number of points with $3$ friends = $20$.

3. Interior Points ($i = 4$ friends):

   • The interior forms a $(7 - 2) \times (7 - 2) = 5 \times 5$ sub-grid.
   • Total interior points = $25$.
   • Each interior point is connected to $4$ adjacent points (left, right, up, down).
   • Number of points with $4$ friends = $25$.

4. Points with $i = 0$ or $i = 1$ friends:

   • There are no isolated or end-degree points in the grid.
   • Number of points with $0$ or $1$ friends = $0$.

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3. Probability Distribution and Expectation Calculation

The probability $p_i = P(X = i)$ for each $i$ is given by: $p_0 = 0$ $p_1 = 0$ $p_2 = \frac{4}{49}$ $p_3 = \frac{20}{49}$ $p_4 = \frac{25}{49}$

The expected value $E(X)$ of the random variable $X$ is: $E(X) = \sum_{i=0}^4 i \cdot p_i$ $E(X) = 0(0) + 1(0) + 2\left(\frac{4}{49}\right) + 3\left(\frac{20}{49}\right) + 4\left(\frac{25}{49}\right)$ $E(X) = \frac{8 + 60 + 100}{49} = \frac{168}{49} = \frac{24}{7}$

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4. Alternative Method (Sum of Degrees)

Alternatively, by graph theory (Handshaking Lemma):

• Total horizontal edges = $7 \times 6 = 42$
• Total vertical edges = $7 \times 6 = 42$
• Total edges $E = 42 + 42 = 84$

The sum of degrees of all 49 points is $2E = 2 \times 84 = 168$.

Thus, the average degree (expected value $E(X)$) is: $E(X) = \frac{\text{Sum of degrees}}{\text{Total points}} = \frac{168}{49} = \frac{24}{7}$

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Conclusion

We need to find the value of $7E(X)$: $7E(X) = 7 \times \frac{24}{7} = 24$