A pair of colors?

Probability Level 3

How many distinct ways can a cube have 2 green sides \color{#20A900} \text{2 green sides} , 2 blue sides \color{#3D99F6} \text{2 blue sides} and 2 red sides \color{#D61F06} \text{2 red sides} ?

Note : Two solutions are only distinct if one cube can't be reoriented to look like the other.

5 3
7
7 1 6 4 2

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Feb 6, 2018

Let the subscript O represent faces of the same color being opposite, and A represent faces (of the same color) being adjacent.

Then you have the following possible setups, with the following number of distinct ways they can be set up:

R O G O B O R_O G_O B_O 1
R O G A B A R_O G_A B_A 1
R A G O B A R_A G_O B_A 1
R A G A B O R_A G_A B_O 1
R A G A B A R_A G_A B_A 2

Total: 6 \boxed6

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