Difference between revisions of "1997 PMWC Problems/Problem T7"
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The corners of the 5-5-5 must be the corners of the 4-4-4, because they have the most colored faces and there are only 8 of them. The edges of the 4-4-4 are the edges of the 5-5-5 minus one, and the faces of the 4-4-4 are the faces of the 5-5-5 minus 9. Then we can fill in the center of the 4-4-4 with anything and it will work. Thus, <math>\boxed{64}</math> works. | The corners of the 5-5-5 must be the corners of the 4-4-4, because they have the most colored faces and there are only 8 of them. The edges of the 4-4-4 are the edges of the 5-5-5 minus one, and the faces of the 4-4-4 are the faces of the 5-5-5 minus 9. Then we can fill in the center of the 4-4-4 with anything and it will work. Thus, <math>\boxed{64}</math> works. | ||
| − | ==See | + | ==See Also== |
{{PMWC box|year=1997|num-b=T6|num-a=T8}} | {{PMWC box|year=1997|num-b=T6|num-a=T8}} | ||
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| + | [[Category:Introductory Combinatorics Problems]] | ||
Revision as of 15:07, 15 May 2012
Problem
Color the surfaces of a cube of dimension 5*5*5 red, and then cut the cube into smaller cubes of dimension 1*1*1. Take out all the smaller cubes which have at least one red surface and fix a cuboid, keeping the surfaces of the cuboid red. Now what is the maximum possible volume of the cuboid?
Solution
The number of cubes with at least one red face is 
. The smallest cube below that is 64, thus we need to prove that we can make a 4-4-4 cube with a totally red surface.
The corners of the 5-5-5 must be the corners of the 4-4-4, because they have the most colored faces and there are only 8 of them. The edges of the 4-4-4 are the edges of the 5-5-5 minus one, and the faces of the 4-4-4 are the faces of the 5-5-5 minus 9. Then we can fill in the center of the 4-4-4 with anything and it will work. Thus, 
works.
See Also
| 1997 PMWC (Problems) | ||
| Preceded by Problem T6 |
Followed by Problem T8 | |
| I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
