2002 AMC 10A Problem 18

Below is the professionally curated solution for Problem 18 of the 2002 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 10A solutions, or check the answer key.

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Concepts:cube geometry optimization casework

Difficulty rating: 1540

18.

A $3\times 3\times 3$ cube is formed by gluing together $27$ standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is $7.$ ) The smallest possible sum of all the numbers showing on the surface of the $3\times 3\times 3$ cube is

$60$

$72$

$84$

$90$

$96$

Solution:

The $8$ corner dice show $3$ faces each, minimized at $1+2+3=6,$ contributing $8\cdot 6=48.$ The $12$ edge dice show $2$ faces, minimized at $1+2=3,$ contributing $12\cdot 3=36.$

The $6$ face-center dice show $1$ face, minimized at $1,$ contributing $6,$ and the hidden interior die contributes $0.$ The total is $48+36+6=90.$

Thus, the correct answer is D.

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