[Math] Instant insanity question

graph theorypuzzle

My question is regarding the necessary conditions that a graph must fulfill to satisfy instant insanity problem.

Now take for example the left, right, front and back face colors of the four cubes be(the colors are red, black, green and white):

Front   Back    Right    Left
R       W         R       G
G       W         W       B
G       B         G       W
B       R         B       R

Now as you can see on front and back, green and white show up twice respectively which violates the problem. But if we make their graph, their graphs fulfill all the conditions:
1.the graph can be dissociated into two edge disjoint subgraphs
2.each subgraph has vertices of degree 2
3.each subgraph has all the edges representing the four cubes once

enter image description here

Note:
I stack the cubes one above the other. This is different from the link i gave in which cubes are kept on the side of each other

Best Answer

I understand you have to play around with the cubes in order to get the answer. You just know what the fronts and backs; and lefts and rights of each cube are going to be.

"To solve the game, the first graph represents the front and back faces, and the second graph represents the top and bottom faces. Align the cubes according to which edges you have in your two graphs. There is some choice on how to do this; for example, on the game above, cube 1 has the front and back faces blue and green, and the top and bottom faces red and yellow. However, we do not yet know whether blue is on the front or back. You may have to play around a little to arrange your cubes appropriately."

I think all of the cubes are fine the way they are except for the second cube. If you play around with the second cube, then you'll see that if you if you flip it (front becomes back, and back becomes front, but right stays right and left stays left) then you have a valid answer that fits your graph.

EDIT: I think I noticed something. If you think of each of the two graphs as a path, in which you can only walk in one direction, then fill in the answers that way. So you went from R to W, W to G, G to B, and B-R for the blue graph. See how that is one continuous path, and you can fill out your chart in that order. Same thing with the orange graph, even if lines 1 and 2 are not connected. Think of the path as going from red to green to white to black back to red. So you would fill out R to G, then W to B, then G to W, and then B to R in your chart. I don't know if this is actually true, but just a theory.

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