A sign manufacturer makes yield signs by cutting N identical-sized equilateral triangles from a square piece of aluminum; N is at most 25. What is the most efficient use of aluminum: what is the optimal value of N and what percent of the aluminum is used in yield signs?

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A sign manufacturer makes yield signs by cutting N identical-sized equilateral triangles from a square piece of aluminum; N is at most 25. What is the most efficient use of aluminum: what is the optimal value of N and what percent of the aluminum is used in yield signs?

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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|dw:1326946628944:dw| optimal value for N is 16 because you get 100% efficiency
those aren't yield signs! (equilateral triangles)
i thought it was isosceles, sorry, my answer is wrong

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http://www2.stetson.edu/~efriedma/triinsqu/
awesome! n = 24 gives an example with better than 86.4% of the metal being used in yield signs. The n=4 example shows almost 70%; the case n=7 is crazy beautiful using equilateral triangles of side length exactly half the square. Since those pictures are repeated in MathWorld and CRC as "best known," I think I'll put this problem into the category "Unlikely to be solved with proof during my lifetime."

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