Consider the cube shown below with its 8 corners labelled a, b, c, d, e, f, g, and h. The figure is representative. All corners are to be colored such that any two corners that are connected by an edge must be of different colors. The minimum number of colors required to achieve this is ________
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Check whether the cube's corner and edge graph contains any cycle of odd length; if every cycle is even in length, the graph can be colored with just 2 colors.
Step 1: Set up the two square faces of the cube. A cube has a top face and a bottom face, each a 4 cornered square, joined by 4 vertical edges. Say the top face has corners a, b, d, c going around it in order (edges a-b, b-d, d-c, c-a), and the bottom face has corners f, h, g, e going around it in the matching order, with vertical edges joining each top corner straight down to the corner directly below it.
Step 2: Try coloring with just 1 color first. With 1 color, corners a and b would both get the same color, but a and b are joined directly by an edge on the top face, which is not allowed. So 1 color fails immediately, meaning at least 2 colors are needed.
Step 3: Attempt a valid coloring using exactly 2 colors by alternating around the cube, corner by corner. Assign Color 1 to corner a. Every corner directly connected to a by an edge must then get Color 2, since it cannot match a. Continue this alternation all the way around: color the corner connected to a Color 2 corner as Color 1 again, and so on, moving around both the top square and down each vertical edge.
Step 4: Verify no contradiction arises while going all the way around the cube. On the top face, going around the 4 corners in a cycle and alternating Color 1, Color 2, Color 1, Color 2 works cleanly because 4 is an even number, so the coloring returns to Color 1 exactly where it started, without any clash. The same alternation carries down each vertical edge to the bottom face, and since the bottom face is also a 4 cornered square (an even length cycle), it also alternates cleanly with no contradiction. Every one of the 12 edges of the cube ends up joining a Color 1 corner to a Color 2 corner.
Step 5: Conclude the minimum. Since a complete, contradiction free coloring was built using only 2 colors, and Step 2 already showed 1 color is impossible, the minimum number of colors needed is 2. Options (A) 8 and (B) 4 use far more colors than necessary, and option (C) 3 is also more than needed since a clean 2-coloring already exists.
