[#52] 2 Min. Challenge: Cube

Question

Time yourself. Please explain your solution. 

A 4 Ã— 4 Ã— 4 cube is made from 32 white unit cubes and 32 black unit cubes. What is the largest possible percent of black surface area?

(A) 50% 
(B) 60% 
(C) 64% 
(D) 66 2/3 % 
(E) 75%

Emmanuel · Answer

There are 4^3 - 2^3 = 56 cubes which are visible on the surface.

32 from black cubes can be allocated among them to maximize black surface area: corner cubes: total # is 56 - 4*6 = 32!!!

So, black should be all corner cubes.

% of surface area in this case: (16 - 4)/16 = 75%.

E is the answer.

anandnk · Answer

This question is not very clear to me. I am going to place the black squares on top and bottom of the cube.

Total surface area of square = 4 * 4 * 6

Total surface area on top A1 = 4*4 + 4*1 * 4
Total surace area on bottom A2 = 4 * 4 + 4*1 * 4
A= A1+A2 = 4* 4 * 4

Ratio = 4 * 4 * 4 / ( 4 * 4 * 6 ) = 66.66

Paul · Answer

haven't worked this one yet but I have to admit that you guys are experts at drawing nice diagrams :good

Paul · Answer

Answer should be 75%
Anandnk, from your top and bottom black cube layers, remove the center 4 cubes and line them up along the sides. That way, you will cover the maximum area with the black cubes since each black cube will show 2 faces.

On each of the 6 faces of the cube, you will have a 2x2 white area in the center. So 6*2*2 = 24 white cube^2.
Total area is 4*4*6 = 96 cube^2
96-24 / 96 = 75%

anandnk · Answer

Yeah I agree. I think I can cover more area by replacing the cubes forming the edge with the center 4 cubes.

Makky07 · Answer

Thanks anandnk, your diagram is a good one and it helped me to easily understand Paul's explanation. Very nice.

Thanks Paul and Emmanuel for your explanations.

Praetorian · Answer

The answer is E.

Put all the black unit cubes at the corners and along the edges
of the big cube so they contribute either 3 or 2 square units to
the surface. There are just enough black unit cubes to fill all eight
corner positions and all the other 12├Ч2 = 24 edge positions, so each
face of the big cube is three-forths covered by black unit squares.

[#52] 2 Min. Challenge: Cube

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