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February 9th, 2015, 06:17 AM  #1 
Member Joined: Jan 2015 From: Orlando, Florida Posts: 92 Thanks: 10  Counting
16 of the 64 unit cubes are painted red on all faces. The 64 unit cubes are then arranged into a 4x4x4 cube. How many arrangements exist such that for every 1x1x4 cube inside the 4x4x4 cube, there is exactly one red cube?

February 9th, 2015, 06:50 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,649 Thanks: 2630 Math Focus: Mainly analysis and algebra 
For an arrangement of red cubes to work for all ways of forming 1x1x4 pieces, no red cube can share a row, column or 'pillar' with another red piece.

February 9th, 2015, 01:21 PM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
Though "1 by 1 by 4" would not be a "cube".


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