Difference between revisions of "1985 AHSME Problems/Problem 20"
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Revision as of 12:01, 5 July 2013
Problem
A wooden cube with edge length 
units (where is an integer 
) is painted black all over. By slices parallel to its faces, the cube is cut into 
smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is ?
Solution
Notice that if we remove the outer layer of unit cubes from the entire cube, we're left with a smaller cube of side length 
. Notice also that this contains all of the unpainted cubes and nothing else, so there are 
unpainted cubes. Also notice that if we take one face of the cube and remove the outer edge, we're left with a square of side length containing all of the cubes on that face with exactly one face painted. This can be done for the other 
faces as well, for a total of 
cubes with only one face painted.
.
See Also
| 1985 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 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. 
