Difference between revisions of "1987 AHSME Problems/Problem 27"
(Created page with "==Problem== A cube of cheese <math>C=\{(x, y, z)| 0 \le x, y, z \le 1\}</math> is cut along the planes <math>x=y, y=z</math> and <math>z=x</math>. How many pieces are there? (N...") |
(Added a solution with explanation) |
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| Line 8: | Line 8: | ||
\textbf{(C)}\ 7 \qquad | \textbf{(C)}\ 7 \qquad | ||
\textbf{(D)}\ 8 \qquad | \textbf{(D)}\ 8 \qquad | ||
| − | \textbf{(E)}\ 9 </math> | + | \textbf{(E)}\ 9 </math> |
| + | |||
| + | == 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 == | ||
Revision as of 13:04, 31 March 2018
Problem
A cube of cheese 
is cut along the planes 
and 
. How many pieces are there?
(No cheese is moved until all three cuts are made.)
Solution
The cut 
separates the cube into points with 
and points with 
, 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 
, 
, and 
- for example, all points with the ordering 
are in the same piece. Thus, as there are 
possible orderings, there are 
pieces, which is answer 
.
See also
| 1987 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 26 |
Followed by Problem 28 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
