1987 AHSME Problems/Problem 27

Problem

A cube of cheese $C=\{(x, y, z)| 0 \le x, y, z \le 1\}$ is cut along the planes $x=y, y=z$ and $z=x$ . How many pieces are there? (No cheese is moved until all three cuts are made.)

$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Solution

The cut $x = y$ separates the cube into points with $x < y$ and points with $x > y$ , and analogous results apply for the other cuts. Thus, which piece a particular point is in depends only on the relative sizes of its coordinates $x$ , $y$ , and $z$ - for example, all points with the ordering $x < y < z$ are in the same piece. Thus, as there are $3! = 6$ possible orderings, there are $6$ pieces, which is answer $\boxed{B}$ .

1987 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 26	Followed by Problem 28
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1987 AHSME Problems/Problem 27

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