Problem 3 · 2024 AMC 8
Medium
Geometry & Measurement
area-decompositiondifference-of-squares
Four squares of side lengths 4, 7, 9, and 10 units are arranged in increasing size order so that their left edges and bottom edges align. The squares alternate in the color pattern white-gray-white-gray, respectively, as shown in the figure. What is the area of the visible gray region, in square units?
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Answer: E — 52 square units.
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Hint 1 of 2
You never see a whole gray square — a smaller one always sits on its bottom-left corner. So what shape is the gray you actually see, and how would you find its area?
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Hint 2 of 2
Technique: each visible gray piece = (its square) − (the square on top). And a2 − b2 = (a+b)(a−b) makes 102−92 = 19 instantly — no squaring.
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Approach: frame = outer square minus the one on top (difference of squares)
- The insight: you never see a whole gray square — a smaller white square always sits on top, leaving only an L-shaped frame. So gray visible = (gray square's area) − (the square covering it).
- Gray 10 under white 9: 102 − 92. Instead of 100 − 81, use a2 − b2 = (a+b)(a−b) = 19×1 = 19 — no big subtraction.
- Gray 7 under white 4: 72 − 42 = (7+4)(7−4) = 11×3 = 33.
- Add the two frames: 19 + 33 = 52. You'll see it again: any time two squares (or any two areas) sit one inside the other, the leftover is their difference — and difference-of-squares makes consecutive sizes like 10 and 9 collapse to just their sum.
Another way — alternating add and subtract (MAA):
- The visible gray is the 10-square minus the 9-square plus the 7-square minus the 4-square: 100 − 81 + 49 − 16.
- = 52.
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