October 26th, 2017, 07:14 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 385 Thanks: 27 Math Focus: Number theory  Block by block
Take a cube built of 125 unit cubes. How many distinct cubes (1x1x1, 2x2x2, 3x3x3, 4x4x4 and 5x5x5) can be made by facing the unit cubes?

October 27th, 2017, 03:54 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
What does "facing the unit cubes" mean?

October 27th, 2017, 04:43 AM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,334 Thanks: 938 
There enough 1x1x1 cubes to make 4 3x3x3 cubes: 125  27  27  27  27 = 17 (leftovers). Is that what you mean? 
October 27th, 2017, 10:28 AM  #4 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 385 Thanks: 27 Math Focus: Number theory 
By "facing" I meant "building into a unique cube." If I started with a 3x3x3 cube, I would have from it 27[1x1x1]+8[2x2x2]+1[3x3x3]=36 unique cubes. The counts are not always cubic {1, 8, 27} like in this example  as a 4x4x4 or greater largest cube would show. A 4x4x4 cube would manifest 100 cubes overall, I believe. 
October 27th, 2017, 10:33 AM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,334 Thanks: 938 
You've lost me...take over and manifest, Country Boy 

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