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The tip-triangle area of an equal-hand clock

The May 21, 2012, edition of “Numberplay,” a puzzle blog on the web site of the New York Times, included the following question: Assume that the three hands of a clock are all the same length. Are the tips of the hands ever the vertices of an equilateral triangle? If not, when are they closest. Here, I’ve interpreted “closest” based on the area of the triangle whose vertices are the hands’ tips. The clock time can be manipulated with sliding controls. The triangle is pictured, and its area is given as a percentage of that of an equilateral triangle.

Contributed by: Steve Kass

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The Numberplay column from the Times is here: http://wordplay.blogs.nytimes.com/2012/05/21/times/.

The area function in this question is not easy to maximize. At 2:54:34.6, the triangle attains 99.9993% of the area of an equilateral triangle. This is the largest value I found.