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 ________
Show Hint
Label each corner by how many of its 3 coordinates are 1, then color by whether that count is even or odd, like a 3D checkerboard. Check whether any single edge of the cube ever joins two corners that share the same parity.
Step 1: Understanding the Concept:
This is a graph coloring problem. The 8 corners of the cube are the vertices of a graph, and the 12 edges of the cube are the edges of that graph. We need the smallest number of colors so that every pair of corners joined by an edge gets different colors; this smallest number is called the chromatic number of the graph.
Step 2: Key Formula or Approach:
Label each corner of the cube with 3 binary coordinates, one bit for each direction \(x\), \(y\), \(z\), so the 8 corners become the 8 strings \(000, 001, 010, 011, 100, 101, 110, 111\). Two corners are joined by a cube edge exactly when their labels differ in exactly one bit, that is, moving along one direction only.
Step 3: Detailed Explanation:
Split the 8 corners into two groups by the parity of the number of 1s in their label: group X, an even count of 1s (\(000, 011, 101, 110\)), and group Y, an odd count of 1s (\(001, 010, 100, 111\)). Since every edge changes exactly one bit, it always changes the count of 1s by exactly 1, which flips even to odd or odd to even. So every edge of the cube joins one corner from group X to one corner from group Y, and never joins two corners from the same group. Color every corner in group X with color 1 and every corner in group Y with color 2, and no edge connects two same-colored corners.
Step 4: Why fewer than 2 colors cannot work.
The cube has at least one edge, and both its ends need different colors, so 1 color is never enough for any graph that has an edge. This means 2 is the smallest possible number of colors, not just a number that happens to work.
Step 5: Why the other options are wrong.
Options (A) 8 and (B) 4 both use more colors than needed. Giving every corner its own separate color, or splitting into 4 colors, both satisfy the different-color-on-every-edge rule, since using more distinct colors than required never breaks a valid coloring, but the question asks for the minimum. Option (C) 3 also gives a valid coloring but is still not minimal, since Step 3 already shows the rule can be met with only 2 colors.
Final Answer:
The cube's corners can be split into exactly two groups so that every edge joins the two groups, so the minimum number of colors needed is 2.
\[ \boxed{2} \]
