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Covering a Cube with Strips (Posted on 2016-08-18) Difficulty: 3 of 5
Show how to cover a size 4 cube with sixteen 1x6 strips. The strips are allowed to bend over an edge to cover multiple faces.

Is it possible to cover a size 5 cube with 25 of the 1x6 strips?

No Solution Yet Submitted by Brian Smith    
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Some Thoughts re: Solution to Part 1. Thoughts on part 2. Extensions. Comment 3 of 3 |
(In reply to Solution to Part 1. Thoughts on part 2. Extensions. by Jer)

Jer writes:

  • If you color the entire cube like a checkerboard and looks at the 150 squares, 78 are black and 72 are white.  Each strip covers 4 of one color and 2 of the other.  This means 14 of the strips are of the 4 black type and 11 are of the 4 white type.  This parity argument seems promising but I'm stuck.
I notice that it is also possible for a strip to cover 3 black and 3 white squares, if it has length 2 on one face and length 4 on another.  But I agree that it is necessary to have 3 more 4 black coverings than 4 white coverings, if the strips are parallel to some of the edges.

Offhand, I do not find a way to make this work, and I suspect that it is not possible.  I even considered coverings where the strips were 45 degrees to the edges, but this seemed unpromising also.

Edited on August 22, 2016, 8:25 am
  Posted by Steve Herman on 2016-08-21 18:35:39

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