2003 AMC 8 Problem 15
Attempt Problem 15 of the 2003 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
15.
A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown?
Answer: B
Small hint:
Try to satisfy both views with only three cubes.
Big hint:
A four-cube construction is shown; you only need to rule out three.
Solution:
The front view requires an L-shape with three visible positions, so at least three cubes are needed.
Suppose there were exactly three cubes. The two cubes that appear one above the other must share a face, so they have the same depth. The third cube appears beside the lower one. To share a face with either of the first two cubes, it must share the lower cube’s depth as well; otherwise it would be disconnected from both. Thus all three cubes would occupy one depth column, giving a one-column side view instead of the required L-shape.
The shown four-cube construction has both required views, so the minimum is
Thus, B is the correct answer.
Problem 15 in other years
- 1985 AMC 8
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