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Unit Cubes (Posted on 2004-02-02) Difficulty: 3 of 5
Some unit cubes are assembled to form a larger cube. Some of the faces of the larger cube are then painted. The cube is taken apart and it is found that 217 of the unit cubes have paint on them. What is the total number of unit cubes?

See The Solution Submitted by Brian Smith    
Rating: 3.8333 (6 votes)

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Short solution | Comment 4 of 15 |
Let P denote the number of painted cubes
P=217 & P <n^3-(n-2)^3
Hence n>8
Let us start with n = 9
Assuming all faces are painted
=> P=729-343=386...
TOO MUCH
Since each unpainted face reduces P by about 49-56 units we try 3 unpainted faces.
They are meeting or not meeting in the corner.
The reduced quantity for the adjacent case is
3*7^2+3*7+1=147+21+1=169
bingo 386-169=217

so the answer is : a total of 9^3=729 cubes,three
adjacent faces painted (blue, I presume).

ady

Edited on February 2, 2004, 7:33 pm

Edited on February 2, 2004, 7:33 pm

Edited on February 2, 2004, 7:55 pm

Edited on February 2, 2004, 7:56 pm

Edited on February 2, 2004, 7:57 pm
Edited on February 2, 2004, 8:00 pm
  Posted by Ady TZIDON on 2004-02-02 16:58:12
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