4. The positive integers from 1 to 576 are written in a $24 \times 24$ grid so that the first row contains the numbers 1 to 24 , the second row contains the numbers 25 to 48 , and so on, as shown. An $8 \times 8$ square is drawn around 64 of these numbers. (These 64 numbers consist of 8 numbers in each of 8 rows.) The sum of the numbers in the four corners of the $8 \times 8$ square is 1646 . What is the $\begin{array}{ccccc}1 & 2 & \cdots & 23 & 24 \\ 25 & 26 & \cdots & 47 & 48\end{array}$ number in the bottom right corner of this $8 \times 8$ square?
4. Let $x$ be the number in the bottom right corner of the $8 \times 8$ square.
Since there are 8 numbers across the bottom row of this square, then the number in the bottom left corner is $x-7$. (The 8 numbers in the bottom row are $x-7, x-6, x-5, x-4, x-3$, $x-2, x-1$, and $x$.)
In the larger grid, each number is 24 greater than the number directly above it.
The number in the bottom right corner of the $8 \times 8$ square is seven rows beneath the number in the upper right corner.
Since there are 24 numbers in each row, each number in the grid above the bottom row is 24 less than the number directly below it.
This means the number in the top right corner is $7 \times 24=168$ less than the number in the
bottom right corner, as it is 7 rows above it.
Since the number in the bottom right corner is $x$, then the number in the top right corner is $x-168$.
Moving to the left across the first row of the $8 \times 8$ square, the number in the top left corner is $(x-168)-7$ or $x-175$.
Since the sum of the numbers in the four corners of the square is to be 1646 , then
$$
\begin{aligned}
x+(x-7)+(x-168)+(x-175) & =1646 \\
4 x-350 & =1646 \\
4 x & =1996 \\
x & =499
\end{aligned}
$$

Therefore, the number in the bottom right corner is 499 .
We should also verify that, starting with 499 in the bottom right corner, we can construct an $8 \times 8$ square that doesn't extend past the top or left border of the grid.
Since the bottom right number in the grid is 576 , then the numbers up the right side of the grid are $576,552,528,504$, which means that, moving from right to left, 499 is sixth number of the 4 th row from the bottom.
Since the grid is $24 \times 24$, then the square does not meet the left or top borders.
ANSWER: 499