Saturday, April 9, 2016

Tetrahedron Coloring Problem

For the solution, click "Read More" below.

The Tetrahedron Coloring Problem

Solution:  #3 is different from the others.  The others all fold into the same tetrahedron.

There are exactly two ways to color a tetrahedron using 4 different colors, and the two colorings differ by a reflection.

Looking at the nets below, we can see that #1 and #5 are really the same tetrahedron because the nets differ only by a rotation.  So the answer must be either 2, 3, or 4.

Macintosh HD:Users:Sian_Zelbo:Google Drive:tetrahedra problem.jpg

To analyze the others, there are different ways to do it.  You may be able to visualize folding them into a 3D figure.  You could even cut them out of paper and fold them to see which ones are different.   

You can also take one of the nets and move the faces in a careful way so that the edges you reconnect are also edges that meet up in the folded 3D figure.  That way you have an equivalent net.  In the diagram below, I started with net #2, and moved two faces to end up with an equivalent net that is a rotation of #5.  Now I can conclude that the "different" one must be either #3 or #4.

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