Difference between revisions of "2025 AMC 8 Problems/Problem 15"
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==Solution 1== | ==Solution 1== | ||
We first observe that there are <math>6^2 - 13 = 23</math> gold squares. To find <math>M</math>, we can note that <math>\frac{23}{2} = 11 \frac{1}{2}</math>, so we can have at most <math>11</math> pairs (which can be constructed). To find <math>m</math>, we fill one side up entirely with gold squares, leaving <math>5</math> left for the other side. From this position, no matter where we move a square, there will always be <math>5</math> overlapping gold-on-gold pairs, so <math>m = 5</math>. It follows that <math>m + M = 5 + 11 = \boxed{\textbf{(C)\ 16}}.</math> | We first observe that there are <math>6^2 - 13 = 23</math> gold squares. To find <math>M</math>, we can note that <math>\frac{23}{2} = 11 \frac{1}{2}</math>, so we can have at most <math>11</math> pairs (which can be constructed). To find <math>m</math>, we fill one side up entirely with gold squares, leaving <math>5</math> left for the other side. From this position, no matter where we move a square, there will always be <math>5</math> overlapping gold-on-gold pairs, so <math>m = 5</math>. It follows that <math>m + M = 5 + 11 = \boxed{\textbf{(C)\ 16}}.</math> | ||
| + | This is another individual. I would like to state that there is a video below if you do not understand the solution. | ||
==Solution 2== | ==Solution 2== | ||
Revision as of 13:55, 30 January 2025
Problem
Kei draws a 
-by- grid. He colors 
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 
and 
equal equal the least and greatest possible number of gold-on-gold pairs, respectively. What is the value of 
?
Solution 1
We first observe that there are 
gold squares. To find , we can note that 
, so we can have at most 
pairs (which can be constructed). To find , we fill one side up entirely with gold squares, leaving 
left for the other side. From this position, no matter where we move a square, there will always be overlapping gold-on-gold pairs, so 
. It follows that 
This is another individual. I would like to state that there is a video below if you do not understand the solution.
Solution 2
First, we note that there are 
"pairs" of squares folded on top of each other after the folding. The minimum number of gold pairs occurs when silver squares occupy the maximum number of pairs, and the maximum number of gold pairs occurs when silver squares occupy the minimum number of pairs. The former case occurs when all silver squares are placed in different pairs, resulting in 
gold pairs. The latter case occurs when the silver squares are paired up as much as possible, resulting in complete pairs and another square occupying another pair slot. Then there are 
gold pairs. Our answer is 
. ~cxsmi
