1998 AMC 8 Problem 21

Below is the professionally curated solution for Problem 21 of the 1998 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1998 AMC 8 solutions, or check the answer key.

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Concepts:complementary counting 3D geometry

Difficulty rating: 1410

21.

A $4\times 4\times 4$ cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch a side or the bottom of the box?

$48$

$52$

$60$

$64$

$80$

Solution:

The only cubes that do not touch a side or the bottom form the interior core above the bottom layer. This core has dimensions $2\times2\times3$ , so it contains $12$ cubes.

Thus $64-12=52$ cubes touch a side or the bottom.

Thus, the correct answer is B .

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