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Using weights of 1, 3, 9, and 27 grams, any number of grams from 0 to 40 can be measured on a double balance scale. Is there a loop through all 41 values, with only a single ternary weight added or removed at each step? No such loop exists. The number of ternary weights alternates between odd and even. Since there are 41 steps, at some point in the loop two steps with an even number of weights must be adjacent. This is an example of a parity argument. For a path, one solution starts 0-1-4-3-2-5, then follows the blue path in the grid graph below. You can explore more with the Balanced Ternary Notation Demonstration at The Wolfram Demonstration Project.
Grid Graph of Ternary Neighbors
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