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Can any one explain to me how to answer this question

Q $a)$In how many ways can vertices of square be $3$ colored if the square can moved in $3$ dimension.

<p>$b)$ Answer the same question but no adjacent vertices to be the same color ? </p>
MR_BD
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leena adam
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1 Answers1

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At first glance the notion of coloring vertices seems unrelated to the property of movement in 3D. However you are likely supposed to consider two colorings equivalent if the colored vertices may be brought into coincidence through rigid motion of the square in 3D. In other words not only the fourfold rotation possible in a plane, but the 3D rotation around an axis lying in the plane that effects a permutation of vertices equivalent to a reflection across that axis.

hardmath
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  • Because the number of colors is not specified, you are probably asked for a formula based on $n$, the number of colors, subject to this symmetry. It must be a fourth degree polynomial, but to find that you need five values. A hand count for $n=4$ seems daunting. – Ross Millikan Apr 22 '13 at 03:53
  • @RossMillikan: If we separate the number of colors used from the number of colors available, the counting should not seem so daunting. The case of one color used is of course easy: multiply the one arrangement by number of colors available. The other extreme, 4 colors used, is also pretty easy: multiply the ways of assigning 4 given colors (4!/8) by the combinations of the available colors taken 4 at a time. The three ways to use 2 different colors and two ways to use 3 different colors are perhaps a little more challenging, but it seems easy enough to do by hand. – hardmath Apr 22 '13 at 13:03