Suppose you’re working on an algebraic expression that involves variables, addition, multiplication, and parentheses. You try repeatedly to expand it using the distributive law. How do you know that the expression won’t continue to expand forever?
For example, expanding
(x + y)(s(u + v) + t)
gives
x(s(u + v) + t) + y(s(u + v) + t),
which has more parentheses than the original expression.
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Set all the variables equal to 2! “The point of the distributive law is that its application doesn’t change the value of the expression,” writes Dartmouth mathematician Peter Winkler. “The value of the initial expression limits the size of anything you can get from it by expansion.”
Winkler received the puzzle from Dick Lipton of the College of Computing at Georgia Tech and included it in his “Seven Puzzles You Think You Must Not Have Heard Correctly” at the seventh Gathering for Gardner, in March 2006.
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