Wednesday, July 8, 2015

Solution to Fourth of July Puzzle: Out of Many, One

Challenge: Form 1 out of the following fractions:
3/5, 1/3, 2/3, 2/3, 2/3, 2/3, 2/3, 3/4.
In other words, create an expression, the value of which is 1, using all of the above fractions together with any or all of the symbols \(+\), \(-\), \(\times\), \(\div\), and parentheses.

My solution was

\(\frac{3}{5}\times\left(\frac{3}{4}-\frac{1}{3}\right)\times\left(\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{2}{3}\right)\div\frac{2}{3} = 1\).

This uses 7 binary operation symbols and two sets of parentheses. Reader jeff's solution was

\( (\frac{2}{3}\div\frac{3}{5})+(\frac{2}{3}\div\frac{3}{4})-(\frac{2}{3}\div\frac{1}{3})+(\frac{2}{3}\div\frac{2}{3}) = 1\).

This solution also uses 7 binary operation symbols, but it's better than mine in the sense since that it requires no parentheses. (The parentheses here add clarity but could be removed leaving a well formed expression equaling 1.)

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