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If a marker is located in one of the corners of a square cube and we begin to draw according to the following conditions, how many edges of the cube can we cross ?

A) You can cross every edge only once.

B) The marker must not be separated from the cube.

C) The marker can only cross the corners and edges of the cube.

here is the image

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  • $\begingroup$ Wait, do you mean we have to stay on the edges, or we can go onto the faces, but just have to pass over an edge? $\endgroup$
    – JonTheMon
    Commented Jan 16, 2015 at 16:32

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Depending on what the question means, the answer is either 9 or 12 edges per previous answers. Here are two diagrams.

Cubes

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  • $\begingroup$ Only edges! You can only cross edges @Len $\endgroup$
    – shayan
    Commented Jan 17, 2015 at 11:35
  • $\begingroup$ The answer is 9 $\endgroup$
    – shayan
    Commented Jan 17, 2015 at 11:38
  • $\begingroup$ @shayan - OK. I would use the word "traverse" instead of "cross" ;-) $\endgroup$
    – Len
    Commented Jan 18, 2015 at 0:35
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The best you can do is 9. Let's number corners as 1-4 on top, 5-8 on bottom (i is connected to i+4). You can do:

(1-2), (2-3), (3-4), (4-1), (1-5), (5-6), (6-7), (7-8), (8-5)

which has 9 edges.

Why is this the best?

As the cube is a graph with 12 edges and 8 vertices and each vertex has a degree of 3. Effectively this means that passing through a vertex "removes" 2 of that degree. The other 1 must be an end of a sequence of vertices and edges. Having 8 "ends" means that you must have 4 sequences of vertices and edges.

As each sequence has at least one edge, by minimizing three of those sequences to a length of one edge, we get that the best we can do is 9 for the fourth sequence.

For a more theoretical approach to that problem have a look at this.

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    $\begingroup$ @shayan It is against the policies puzzling.SE to ask questions from ongoing contests $\endgroup$
    – dmg
    Commented Jan 16, 2015 at 16:49
  • $\begingroup$ oh i didnt know delete your comment plz i dont do this any more :( $\endgroup$
    – shayan
    Commented Jan 16, 2015 at 16:51
  • $\begingroup$ he he ! i just wanted you to be more serious! its not a contest me and my friends are playing! :| $\endgroup$
    – shayan
    Commented Jan 16, 2015 at 16:53
  • $\begingroup$ the xperia c3 will be for the winner for 1 week:| its not a real contest man...:| $\endgroup$
    – shayan
    Commented Jan 16, 2015 at 16:53
  • $\begingroup$ so you were saying... are you sure its right? $\endgroup$
    – shayan
    Commented Jan 16, 2015 at 16:54
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I think you can hit all 12 edges. Path (faces) is: top-left-back-top-right-back-bottom-right-front-bottom-left-front-top

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  • $\begingroup$ i dont think so cuz i said: you can cross from each edge only once. $\endgroup$
    – shayan
    Commented Jan 16, 2015 at 16:47

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