Kei draws a 6-by-6 grid. He colors 13 of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let $m$ and $M$ equal the least and greatest possible number of gold-on-gold pairs, respectively. What is the value of $m + M$?

A. 12

B. 14

C. 16

D. 18

E. 20

1. 2025 AMC 8 官方解析

Answer (C): There are 13 silver squares and $6^2 - 13 = 23$ gold squares. When the grid is folded in half, there are 18 pairs of squares, each of which can be silver-on-silver, silver-on-gold, or gold-on-gold.

There cannot be 12 gold-on-gold pairs because that would require at least 24 gold squares. The figure below shows it is possible to have 11 gold-on-gold pairs, so the greatest possible number of gold-on-gold pairs is $M = 11$.

Similarly, the combined number of silver-on-silver pairs and silver-on-gold pairs is at most 13 because there are only 13 silver squares. This means the least possible number of gold-on-gold pairs is $m = 18 - 13 = 5$. The figure below shows this can also be achieved.

Therefore, the value of $m + M$ is $5 + 11 = 16$.