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

Question

Pointing to a photograph, a man says, "I have no brothers or sisters, but the father of the person in the photograph is my father’s son." Who is the person in the photograph?

Answer 1

Think of the cube corners as points with coordinates (x, y, z), where each of x, y, z is either 0 or 1. That gives exactly 8 points, one for each corner a through h. Two corners are joined by an edge of the cube exactly when their coordinates differ in exactly one place, for example (0,0,0) and (1,0,0) differ only in the x value.

Coloring rule by parity: color each corner using the sum of its three coordinates, $x+y+z$. If the sum is even, use color X; if the sum is odd, use color Y.
Why this works: moving along a single edge changes exactly one coordinate by 1, so the sum $x+y+z$ always flips from even to odd or odd to even. That means the two corners at the ends of any edge always land in different color groups.
Checking it covers every edge: since every one of the cube's 12 edges changes only one coordinate, this parity argument applies to all of them at once, not just a few sample edges.
Why fewer colors will not work: with a single color, every edge would connect two same colored corners, which the question rules out. So 1 color is impossible, while 3, 4, or 8 colors are all more than what is needed.

This coordinate argument shows the cube's corners split cleanly into two color classes, even-sum corners and odd-sum corners, and no edge ever sits inside one class. So 2 colors are both necessary and sufficient.

Let's summarize:

The cube's corners can be split into two groups so that every edge goes between the groups, never within one.
Coloring each group a single color satisfies the different-color-across-every-edge rule.

The minimum number of colors needed is 2, matching option (D).

Answer 2

Think of the cube corners as points with coordinates (x, y, z), where each of x, y, z is either 0 or 1. That gives exactly 8 points, one for each corner a through h. Two corners are joined by an edge of the cube exactly when their coordinates differ in exactly one place, for example (0,0,0) and (1,0,0) differ only in the x value.

Coloring rule by parity: color each corner using the sum of its three coordinates, $x+y+z$. If the sum is even, use color X; if the sum is odd, use color Y.
Why this works: moving along a single edge changes exactly one coordinate by 1, so the sum $x+y+z$ always flips from even to odd or odd to even. That means the two corners at the ends of any edge always land in different color groups.
Checking it covers every edge: since every one of the cube's 12 edges changes only one coordinate, this parity argument applies to all of them at once, not just a few sample edges.
Why fewer colors will not work: with a single color, every edge would connect two same colored corners, which the question rules out. So 1 color is impossible, while 3, 4, or 8 colors are all more than what is needed.

This coordinate argument shows the cube's corners split cleanly into two color classes, even-sum corners and odd-sum corners, and no edge ever sits inside one class. So 2 colors are both necessary and sufficient.

Let's summarize:

The cube's corners can be split into two groups so that every edge goes between the groups, never within one.
Coloring each group a single color satisfies the different-color-across-every-edge rule.

The minimum number of colors needed is 2, matching option (D).

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

The Correct Option is D

Solution and Explanation

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