Difference between revisions of "1987 AHSME Problems/Problem 27"

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\textbf{(C)}\ 7 \qquad
 
\textbf{(C)}\ 7 \qquad
 
\textbf{(D)}\ 8 \qquad
 
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9    </math>  
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\textbf{(E)}\ 9    </math>
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== Solution ==
 +
The cut <math>x = y</math> separates the cube into points with <math>x < y</math> and points with <math>x > y</math>, 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 <math>x</math>, <math>y</math>, and <math>z</math> - for example, all points with the ordering <math>x < y < z</math> are in the same piece. Thus, as there are <math>3! = 6</math> possible orderings, there are <math>6</math> pieces, which is answer <math>\boxed{B}</math>.
  
 
== See also ==
 
== See also ==

Latest revision as of 17:45, 7 December 2020

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 $\boxed{B}$.

See also

1987 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 26
Followed by
Problem 28
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