Problem 14 · AMC 8 Stretch
Core
Geometry & Measurement
Counting & ProbabilityLogic & Word Problems
asking-key-questionsseeking-complementssymmetry
Ten checkers are set up in a triangle pointing UP, with rows of \(1, 2, 3,\) and \(4\). What is the fewest checkers you must move so the triangle points DOWN instead?
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Answer: 3 checkers
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Hint 1 of 3
Instead of asking which checkers to move, ask the opposite (complement) question: which checkers can STAY where they are because they're already in the right spot for the downward triangle too?
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Hint 2 of 3
Lay the downward triangle on top of the upward one (same overall outline, flipped). Many checkers overlap — those don't need to move.
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Hint 3 of 3
Count how many checkers sit in the overlap and can stay. The answer is \(10\) minus that number.
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Approach: Seeking complements — count the checkers that can stay
- Ask the complement question: how many checkers can stay put? A checker can stay if its spot is part of BOTH the up-triangle and the down-triangle.
- When you overlay the upward triangle and its upside-down version, \(7\) of the \(10\) checkers land on shared spots and don't have to move. Only \(3\) checkers are out of place — the single checker at the top point and the two checkers at the bottom corners.
- So the fewest moves is \(3\).
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