Question 1196587: Twenty seven unit cubes are each coloured completely blue or completely yellow. All twenty seven unit cubes are assembled into a larger cube. If half of the surface area of the larger cube is blue. What is the largest number of unit cubes, that could have been coloured blue
Answer by ikleyn(52781)   (Show Source):
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Twenty seven unit cubes are each coloured completely blue or completely yellow.
All twenty seven unit cubes are assembled into a larger cube.
If half of the surface area of the larger cube is blue.
What is the largest number of unit cubes, that could have been coloured blue
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The large cube, assembled of 27 small cubes, has the edge length of 3 units.
Its surface area is  = 6*9 = 54 square units.
Half of the surface area is 54/2 = 27 square units.
So, from the problem, there are 27 small blue squares at the surface of the large cube.
Also, there are 27 small yellow squares at the surface of the large cube.
The number of blue cubes will be maximum, when the number of yellow cubes under given conditions will be minimum.
So, we place yellow cubes there, where they can show maximum number of their faces.
Such places are the corner cubes: they show 3 faces each.
There are 8 small corner cubes.
8 yellow corner cubes show 3*8 = 24 faces.
So far, 27-24 = 3 yellow faces/cubes remain unidentified yet.
Next, consider small cubes in the middle of beach edge.
The number of edges is 12, and each such small cube shows 2 faces.
We can place 1 yellow cube there - it will provide exactly 2 yellow faces from remaining 3.
Finally, we can place one yellow face/surface at the center of some big 3x3-face.
It gives 8 corner yellow cubes + 1 yellow cube at the middle of some of 12 edges + 1 yellow cube at the center of some 3x3 face
8 + 1 + 1 = 10 yellow cubes.
Placing this way, we provide minimum number of yellow cubes, which is 10.
The rest of cubes, 27 - 10 = 17 are blue cubes.
ANSWER. Maximum possible number of blue cubes is 17.
Solved.
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