document.write( "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
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Algebra.Com's Answer #829487 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Twenty seven unit cubes are each coloured completely blue or completely yellow.
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document.write( "The large cube, assembled of 27 small cubes, has the edge length of 3 units.\r\n" );
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document.write( "Its surface area is   = 6*9 = 54 square units.\r\n" );
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document.write( "Half of the surface area is 54/2 = 27 square units.\r\n" );
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document.write( "So, from the problem, there are 27 small blue squares at the surface of the large cube.\r\n" );
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document.write( "Also, there are 27 small yellow squares at the surface of the large cube.\r\n" );
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document.write( "The number of blue cubes will be maximum, when the number of yellow cubes under given conditions will be minimum.\r\n" );
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document.write( "So, we place yellow cubes there, where they can show maximum number of their faces.\r\n" );
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document.write( "Such places are the corner cubes: they show 3 faces each.\r\n" );
document.write( "There are 8 small corner cubes.\r\n" );
document.write( "8 yellow corner cubes show 3*8 = 24 faces.\r\n" );
document.write( "So far, 27-24 = 3 yellow faces/cubes remain unidentified yet.\r\n" );
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document.write( "Next, consider small cubes in the middle of beach edge.\r\n" );
document.write( "The number of edges is 12, and each such small cube shows 2 faces.\r\n" );
document.write( "We can place 1 yellow cube there - it will provide exactly 2 yellow faces from remaining 3.\r\n" );
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document.write( "Finally, we can place one yellow face/surface at the center of some big 3x3-face.\r\n" );
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document.write( "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\r\n" );
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document.write( "    8 + 1 + 1 = 10 yellow cubes.\r\n" );
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document.write( "Placing this way, we provide minimum number of yellow cubes, which is 10.\r\n" );
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document.write( "The rest of cubes, 27 - 10 = 17 are blue cubes.\r\n" );
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document.write( "ANSWER.  Maximum possible number of blue cubes is 17.\r\n" );
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