Competition problem — Combinatorics

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Combinatorics Difficulty 4.9 AIME Find the answer

3. A large cube is formed by $8 \times 8 \times 8$ unit cubes. How many lines can pass through the centers of 8 unit cubes?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

(Prompt: Add another layer of unit cubes on the outside, prove that the line is in one-to-one correspondence with the added unit cubes. Answer: $\left.\frac{1}{2}\left(10^{3}-8^{3}\right)\right)$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.