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?

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**