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A348668
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Decimal expansion of the probability that a triangle formed by three points uniformly and independently chosen at random in a rectangle with dimensions 1 X 2 is obtuse.
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0
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7, 9, 8, 3, 7, 4, 2, 8, 5, 1, 2, 6, 9, 2, 1, 0, 6, 0, 3, 8, 5, 1, 0, 4, 7, 9, 4, 1, 8, 7, 3, 5, 8, 7, 5, 2, 2, 8, 6, 3, 1, 6, 5, 8, 3, 0, 2, 0, 5, 0, 9, 4, 1, 1, 0, 1, 8, 9, 2, 4, 4, 6, 9, 7, 0, 2, 8, 8, 4, 0, 5, 3, 9, 5, 2, 8, 3, 8, 7, 3, 1, 3, 8, 5, 4, 2, 8, 9, 5, 8, 3, 6, 8, 1, 6, 1, 4, 1, 5, 7, 2, 7, 1, 0, 2
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OFFSET
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0,1
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COMMENTS
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The problem of calculating this probability was proposed by Hawthorne (1955) and solved by Langford (1969, 1970). It was mentioned as an unsolved problem in Ogilvy (1962).
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REFERENCES
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A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, pp. 250-253.
Paul J. Nahin, Digital Dice: Computational Solutions to Practical Probability Problems, Princeton University Press, 2008, pp. 8-11.
Luis A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley, 1976, pp. 21-22.
C. Stanley Ogilvy, Tomorrow's Math: Unsolved Problems for the Amateur, Oxford University Press, New York, 1962, p. 114.
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LINKS
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Frank Hawthorne, Problem E1150, The American Mathematical Monthly, Vol. 62, No. 1 (1955), p. 40; Obtuse triangle within a rectangle, Solution to Problem E1150, ibid., Vol. 78, No. 4 (1971), p. 405.
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FORMULA
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Equals 1199/1200 + 13*Pi/128 - 3*log(2)/4.
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EXAMPLE
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0.79837428512692106038510479418735875228631658302050...
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MATHEMATICA
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RealDigits[1199/1200 + 13*Pi/128 - 3*Log[2]/4, 10, 100][[1]]
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PROG
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(PARI) 1199/1200 + 13*Pi/128 - 3*log(2)/4 \\ Michel Marcus, Oct 29 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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