2025 AMC 8 Problems/Problem 15
Contents
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
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
Solution 2
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 I would like to state that there is a video below if you do not understand the solution.
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=jTTcscvcQmI You can contact him through his Youtube Channel
Video Solution by Thinking Feet
See Also
2025 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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All AJHSME/AMC 8 Problems and Solutions |
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