Difference between revisions of "1998 AHSME Problems/Problem 27"
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== See also == | == See also == | ||
| + | [http://mathworld.wolfram.com/MengerSponge.html Menger Sponge] | ||
{{AHSME box|year=1998|num-b=26|num-a=28}} | {{AHSME box|year=1998|num-b=26|num-a=28}} | ||
[[Category:Intermediate Combinatorics Problems]] | [[Category:Intermediate Combinatorics Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 20:02, 18 December 2017
Problem
A 
cube is composed of twenty-seven 
cubes. The big cube is ‘tunneled’ as follows: First, the six cubes which make up the center of each face as well as the center cube are removed. Second, each of the twenty remaining cubes is diminished in the same way. That is, the center facial unit cubes as well as each center cube are removed. The surface area of the final figure is:
Solution
Solution 1
Each cube has eight faces on each side, for a surface area of 
on the outside. Each face also has to count the surface area in the inside area of the removed cube, for an additional surface area of 
. Thus the total surface area for each is 
.
There are 
of these cubes, for an area of 
. However, many of the cubes share a common face; each corner 
cube has three hidden faces and each edge cube has two hidden faces, for a total of 
hidden faces. Each hidden face has a surface area of 
, so the surface area of the final figure is 
.
Solution 2
After the first step, twenty cubes remain, with corner cubes and 
edge cubes. Each one of these corner cubes contributes 
square units of area, and each edge cube contributes 
square units of area.
The second stage takes away 
square units of area (
for each exposed face) from each of the eight corner cubes, and adds an additional 
more units from the center facial cubes removed. Similarly, the twelve edge cubes each lose 
square nits but gain units. Thus, the total surface area is
![\[8 \cdot (27 - 3 + 24) + 12 \cdot (36 - 4 + 24) = 1056\]](http://latex.artofproblemsolving.com/2/5/f/25f96bb848fdb3d17ea9c6485585084cbb47cd23.png)
See also
| 1998 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 | ||
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