Question

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 $m+M?$

$12$

$14$

$16$

$18$

$20$

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Written solution:

The number of gold squares is $6 \times 6 - 13 = 36 - 13 = 23.$ The $36$ total squares overlap as $18$ pairs.

To minimize the number of pairs with two gold squares, the gold squares should first be spread out across all pairs. That uses up $18$ of them. The remaining $23 - 18 = 5$ gold squares double-up and create a total of $m = 5$ gold-on-gold pairs.

To maximize the number of pairs with two gold squares, the $23$ gold squares should first be paired up as much as possible. That can be done to create $M = 11$ pairs, with $1$ gold square left over, because $23 \div 2$ is $11$ with a remainder of $1.$

The answer is $m + M = 5 + 11 = 16,$ which is choice C.

2025 AMC 8 Problem 15

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Problem 15 in Other Years