Ternary Marbles
You, Ron, Pat, and Will play a game with a jar full of red, pink, and white marbles.
If Ron is given two marbles, he will keep the reddest marble and pass on the other marble. For example, if he is given a white and pink marble, he will keep the pink marble (as the "reddest" marble) and pass on the white marble.
If Will is given two marbles, he will keep the whitest marble and pass on the other marble. For example, if he is given a pink and red marble, he will keep the pink marble (as the "whitest" marble) and pass on the red marble.
If Pat is given any of marble, she will exchange it for a marble with an opposite red and white color values and pass on this new marble. For example, if she is given a red marble, she will pass on a white marble, or if she is given a pink marble, she will pass on a pink marble, or if she is given a white marble, she will pass on a red marble.
If you pick any two marbles out of the jar, and give them to Pat to exchange, who then gives them to Ron to keep one and pass the other to you (for a sequence of: You, Pat, Ron, You), which other sequence would give the same ending marble as when using the same two beginning marbles?
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1 solution
Using the given information, tables can be set up for Pat, Ron, and Will, where the possible starting marble combinations are in the left column(s), and their output is in the right column:
Using the information above, we can find the different outputs for each of the 6 possible starting marble combinations, and find that the (You, Pat, Ron, You) sequence gives the same outputs as the (You, Will, Pat, You) sequence:
Note that if we ignore pink and let red = true and white = false, then Pat's table is equivalent to a logical negation, Ron's table is equivalent to a logical conjunction, and Will's table is equivalent to a logical disjunction. Defining pink as above shows that De Morgan's Law not only applies to binary values but can apply to ternary values as well.