Problem 1360

1. Understanding the Problem:
Farmer John has a $47 \times 53$ rectangular grid. He plants corn on any square of the form $47x + 53y$ for non-negative integers $x$ and $y$. We need to find the perimeter of the contiguous region $R$ formed by the unplanted squares.

2. Expressibility:
A number is expressible if it can be written in the form $47a + 53b$ for non-negative integers $a$ and $b$. We need to determine which numbers are not expressible.

3. Chicken McNugget Theorem:
The Chicken McNugget theorem states that for two coprime integers $m$ and $n$, the largest number that cannot be expressed as $am + bn$ (where $a$ and $b$ are non-negative integers) is $mn - m - n$. Here, $m = 47$ and $n = 53$, which are coprime.

Therefore, the largest number that cannot be expressed is:
$$47 \cdot 53 - 47 - 53 = 2491 - 100 = 2391$$

4. Non-expressible Numbers:
All numbers from $1$ to $2391$ that cannot be expressed as $47a + 53b$ are unplanted. However, since we are dealing with a grid, we need to consider the numbers modulo $47$ and $53$.

5. Column-wise Analysis:
We analyze the grid column-wise. Each column $c$ (where $1 \leq c \leq 46$) contains numbers of the form $c + 47k$ for $k = 0, 1, 2, \ldots$. We need to determine when $c + 47k$ is expressible.

6. Modulo Condition:
Suppose $c + 47k$ is expressible. Then, there exist non-negative integers $a$ and $b$ such that:
$$c + 47k = 47a + 53b$$
Rearranging, we get:
$$c = 47(a - k) + 53b$$
This implies $c \equiv 53b \pmod{47}$. Since $53 \equiv 6 \pmod{47}$, we have:
$$c \equiv 6b \pmod{47}$$
Therefore, $b \equiv \frac{c}{6} \pmod{47}$.

7. Unplanted Squares in Each Column:
For each column $c$, the number of unplanted squares is $9c \pmod{53}$. We need to count the unplanted squares in each column and determine the perimeter of the contiguous region $R$.

8. Perimeter Calculation:
The unplanted squares form 8 "staircases" connected edge-to-edge. Each staircase has a maximum height and width. The perimeter of each staircase is $2(h + w)$.

Summing up the perimeters of all staircases:
$$2(5+45) + 2(6+46) + 2(6+47) + 2(6+48) + 2(6+49) + 2(6+50) + 2(6+51) + 2(5+43) = 850$$

9. Overcounting Correction:
We overcount twice the edges where the staircases are connected. The total length of these edges is $1 + 2 + \dots + 7 = 28$. Therefore, we subtract $2 \cdot 28$ from the total perimeter.

Final perimeter:
$$850 - 2 \cdot 28 = 850 - 56 = 794$$

The final answer is $\boxed{794}$.