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A. In the file cubes.pdf are
two diagrams that can be cut out and folded into a 1x1x1 and a 2x2x2
cube. In the figure at the right is shown a 3x3x3 cube (the famous
"Rubik's cube"). Using a single edge, a single face, and
the volume of the 1x1x1 cube as "1", figure out the edge
length (
In counting the edge length of the larger cubes, only count the outer edges, not the lines making criss-crosses across each face of the cube. Given that
a 0x0x0 cube has edge length, area, and volume equal to 0, graph |

B. If you had a cube of side *N*, where *N* is any integer, what
would be the total edge length (*L*), the total surface area (*A*),
and the total volume (*V*). Give an equation expressing your answer in
terms of the symbol "*N*". Explain how you got your equations.

C. According to your equations, for what value of *N* does *L* = *A*? For what
value of *N* does *L* = *V*?

D. Suppose you got the value of N in the parts above by measuring your cube
in inches. Now convert to centimeters. If the edge of the cube in inches is *N*,
what is it in cm? Now what is *L*? *A*? *V*?

E. Even though you found an *N* for which *L* = *A* when *N* was counted in inches,
if it's counted in cm, you should have found that the equality no longer holds.
What's going on? Did you make a mistake? If so, explain what it is. Or is the
question fundamentally flawed? If it is, explain what's going on.

Problem by Ben Dryefus and E. F. Redish

Page last modified September 4, 2010: G43