1998 AMC 12 Problem 27
Attempt Problem 27 of the 1998 AMC 12 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 1998 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
27.
A cube is composed of twenty-seven cubes. The big cube is “tunneled” as follows: First, the six cubes which make up the center of each face as well as the center cube are removed as shown. Second, each of the twenty remaining cubes is diminished in the same way. That is, the center facial unit cubes as well as each center cube are removed. The surface area of the final figure is
Answer: E
Small hint:
After the first stage, classify the twenty remaining large subcubes as corner or edge cubes
Big hint:
For each second-stage tunnel, subtract exposed center squares and add the newly exposed tunnel walls
Solution:
After the first stage, corner subcubes contribute exposed units each, and edge subcubes contribute each. Tunneling a corner subcube removes exposed unit squares and adds tunnel-wall squares; tunneling an edge subcube removes and also adds Hence the final area is Thus E is correct.
Problem 27 in other years
- 1950 AMC 12
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- 1952 AMC 12
- 1953 AMC 12
- 1954 AMC 12
- 1955 AMC 12
- 1956 AMC 12
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