# Marker and the cube

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.

• 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? – JonTheMon Jan 16 '15 at 16:32

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

• Only edges! You can only cross edges @Len – shayan Jan 17 '15 at 11:35
• The answer is 9 – shayan Jan 17 '15 at 11:38
• @shayan - OK. I would use the word "traverse" instead of "cross" ;-) – Len Jan 18 '15 at 0:35

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.

• @shayan It is against the policies puzzling.SE to ask questions from ongoing contests – dmg Jan 16 '15 at 16:49
• oh i didnt know delete your comment plz i dont do this any more :( – shayan Jan 16 '15 at 16:51
• he he ! i just wanted you to be more serious! its not a contest me and my friends are playing! :| – shayan Jan 16 '15 at 16:53
• the xperia c3 will be for the winner for 1 week:| its not a real contest man...:| – shayan Jan 16 '15 at 16:53
• so you were saying... are you sure its right? – shayan Jan 16 '15 at 16:54

I think you can hit all 12 edges. Path (faces) is: top-left-back-top-right-back-bottom-right-front-bottom-left-front-top

• i dont think so cuz i said: you can cross from each edge only once. – shayan Jan 16 '15 at 16:47