logo
Interesting Math Problem   Math Problems Home  |  Home  |  Send Feedback

Cutting up a cube

Tim has a solid wooden cube with whole number dimensions. He paints the
entire surface of the cube red. Then, with slices parallel to the faces
of the cube, Tim cuts the cube into 1x1x1 cubes. A certain number of the
small cubes are completely free of paint (x). A certain number of the
small cubes are painted red on only one side (y). A certain number of
the small cubes are painted red on two sides (z).

A) If y is twice as big as x, what was Tim's original cube size ?
B) If x is twice as big as y, what was Tim's original cube size ?
C) If y + z is 33% of x, what was Tim's original cube size ?

We'll call the 1x1x1 cubes "cubies".
Let the dimension of the cube be n.
Total number of cubies is n³.

There will be 8 corner cubies with 3 sides painted.
x = (n-2)³, the non-painted cubies.
y = (n-2)² x 6, the non-edge, on the face cubies.
z = (n-2) x 12, the edges (not corners) of the cube.

A) If y is twice as big as x, what was Tim's original cube size ?
y = 2x
2 (n-2)³ = 6 (n-2)²
n-2 = 3
n = 5 (x = 27, y = 54)

B) If x is twice as big as y, what was Tim's original cube size ?
B) x = 2y
(n-2)³ = 12 (n-2)²
n-2 = 12
n = 14 (x = 1728, y = 864)

C) If y + z is 33% of x, what was Tim's original cube size?
y + z = .33 x
Let m = (n-2)
6 m² + 12 m = .33 m³
6 m + 12 = .33 m²
2m + 4 = .11 m²
.11 m² - 2m - 4 = 0

2m = 2 + sqrt ( 4 + 4 * 4 * .11 )
.22m = 2 + sqrt ( 4 + 1.76 )
.22m = 2 + sqrt (5.76)
.22m = 2 + 2.4 = 4.4
22 m = 440
m = 20
n = 22 (x = 8000, y = 2400, z = 240)