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Problem of the Day #38: You will be baked (and then there will be) cake April 26, 2011

Posted by Saketh in : potd , trackback

Mitchell is decorating a cube-shaped cake using his collection of icing pens. The design he has in mind requires three colors on each face, as shown in the diagram below:

Mitchell would like to add some color to his cake, and has $6$ different icing pens available for this purpose. Although the diagram shows a cube with the same coloring on all of its faces, he is going to color each face of his cake independently. He thus has a total of $18$ different color choices to make while icing the cake.

How many distinct colorings of Mitchell’s cake are possble? Two colorings are not considered distinct if one can be rotated to obtain the other. The orientation of the hearts on the faces of the cube is negligible.

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