Triangle Mysteries

This game starts with a row of squares of three different colors. The rule for choosing the color for a square in the next row is:
If the two colors above the square are different, then it is colored with the third color; if the two colors above are the same, then the square is colored the same.
Apply this rule repeatedly until there is only one square at the bottom.
The object of this game is to answer the following question: can you predict, just by looking at the top row, what the color of the bottom square will be? That is, can you devise a rule for determining the color of the bottom square that only involves knowing the colors of the first row?
Hint: a rule exists for and another for , and they turn out to be the same. For the whole story, see [1].


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[1] E. Behrends and S. Humble, "Triangle Mysteries," The Mathematical Intelligencer, 35(2), 2013 pp. 10–15.
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