No colorful points

Probability Level 3

To the right is a dodecahedron, a 12-sided Platonic solid .

Your challenge is to paint all the faces, and each face can be $\color{#D61F06} \text{red}$ , $\color{#20A900} \text{green}$ or $\color{#3D99F6} \text{blue}$ .

If you paint $\color{#D61F06} r \text{ sides red}$ , $\color{#20A900} g \text{ sides green}$ , and $\color{#3D99F6} b \text{ sides blue}$ , what is the maximum value you can get for $\color{#D61F06}r \color{#3D99F6} b \color{#20A900} g$ --the product of all three--given that no three colors can meet at a vertex?

The answer is 54.

1 solution

Geoff Pilling
Feb 5, 2018

The only way you can not have three colors meeting at a vertex, is if two colors are completely separated from each other by the third color, forming their own "islands", and the maximum of the product $rbg$ is realized when these two islands are similarly sized and as large as possible. So, the solution is when you have three like colors meeting at one vertex, three of a different color meeting at the opposite vertex, and the remaining faces (6 of them) filled in with the third color.

$6 \times 3 \times 3 = \boxed{54}$

No colorful points

The answer is 54.

1 solution

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