, 2023

A grid with 12 rows and 15 columns has $12\times15=180$ pieces. Solution 1 We begin by recognizing that the middle pieces in each grid form a rectangle. In a grid with 6 rows, the 1st row and the 6th row are each composed entirely of edge pieces, and thus the grid has $6-2=4$ rows that contain some middle pieces. In each of these 4 rows, the 1st column and the 4th column are each composed entirely of edge pieces, and thus the grid has $4-2=2$ columns that contain some middle pieces. Therefore, a grid with 6 rows and 4 columns contains a rectangular grid of middle pieces having 4 rows and 2 columns, and thus has $4\times2=8$ middle pieces. Solution 2 A grid with 6 rows and 4 columns has $6\times4=24$ pieces. We proceed to find the number of edge pieces, and then subtract this number from 24 to determine the number of middle pieces. The first column of the grid contains 6 edge pieces (since there are 6 rows), and the fourth column of the grid also contains 6 edge pieces. The first row of the grid contains 4 edge pieces (since there are 4 columns). However, the first and last of these edge pieces (the top left and right corners of the grid) were previously included in the count of edge pieces in the first and last columns, respectively, and so there are $4-2=2$ additional edge pieces in the first row. Similarly, there are 2 additional edge pieces in the sixth row. Thus, there are $6+6+2+2=16$ edge pieces, and so there are $24-16=8$ middle pieces. Since 14 has two possible factor pairs, 1 and 14 or 2 and 7, then the dimensions of the rectangular grid of middle pieces has either 1 row and 14 columns (or vice versa), or it has 2 rows and 7 columns (or vice versa). If the rectangular grid of middle pieces has 1 row, then the puzzle grid has $1+2=3$ rows since there is a row of edge pieces both above and below the 1 row of middle pieces. Similarly, if the rectangular grid of middle pieces has 14 columns, then the puzzle grid has $14+2=16$ columns since there is a column of edge pieces both to the right and to the left of the middle pieces. In this case, the puzzle grid has 3 rows and 16 columns (or vice versa), and thus has $3\times16=48$ pieces. A puzzle grid with 48 pieces, including 14 middle pieces, has $48-14=34$ edge pieces. If the rectangular grid of middle pieces has 2 rows, then the puzzle grid has $2+2=4$ rows, as above. Similarly, if the rectangular grid of middle pieces has 7 columns, then the puzzle grid has $7+2=9$ columns. In this case, the puzzle grid has 4 rows and 9 columns (or vice versa), and thus has $4\times9=36$ pieces. A puzzle grid with 36 pieces, including 14 middle pieces, has $36-14=22$ edge pieces. The values of $s$ and $t$ are 34 and 22. A grid with 5 rows and $c$ columns contains $5c$ pieces. A grid with 5 rows and $c$ columns contains a rectangular grid of middle pieces with $5-2=3$ rows and $c-2$ columns, and thus has $3(c-2)$ middle pieces. Since the number of edge pieces is equal to the number of middle pieces, then the total number of pieces is twice the number of middle pieces. Thus, $5c=2\times3(c-2)$ or $5c=6(c-2)$. Solving, we get $5c=6c-12$ and so $c=12$.