【Question 16】
A chessboard has 13 rows and 17 columns, with each small square containing a number. Starting from the top left corner, the first row is sequentially $1,2, \cdots, 17$; the second row is sequentially $18,19, \cdots, 34 ; \cdots$ until the last row. Now, the numbers on this chessboard are rewritten, starting from the top left corner, the first column from top to bottom is sequentially $1,2, \cdots, 13$; the second column from top to bottom is sequentially $14,15, \cdots, 26 ; \cdots$ until the last column. In this way, some small squares have the same number in both writing methods, then the sum of all these small squares (with the same number) is $\qquad$
【Analysis and Solution】
Indeterminate equation.
Let the same number in the two different representations be in the $x$-th row and $y$-th column $(1 \leqslant x \leqslant 13, 1 \leqslant y \leqslant 17)$;
This number $=(x-1) \times 17 + y = (y-1) \times 13 + x$;
Simplifying, we get $4 x - 3 y = 1$;
Solving, we get $\left\{\begin{array}{l}x=1 \\ y=1\end{array}\right.$ or $\left\{\begin{array}{l}x=4 \\ y=5\end{array}\right.$ or $\left\{\begin{array}{l}x=7 \\ y=9\end{array}\right.$ or $\left\{\begin{array}{l}x=10 \\ y=13\end{array}\right.$ or $\left\{\begin{array}{l}x=13 \\ y=17\end{array}\right.$;
There are 5 numbers that are the same in the two different representations: $1, 56, 111, 166, 221$; the sum of all these numbers in the small squares (with the same number) is $1 + 56 + 111 + 166 + 221 = 555$.