A348668 - OEIS

A348668

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.

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

0,1

COMMENTS

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).

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..104.

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.

Eric Langford, The probability that a random triangle is obtuse, Biometrika, Vol. 56, No. 3 (1969), p. 689.

Eric Langford, A problem in geometric probability, Mathematics Magazine, Vol 43, No. 5 (1970), pp. 237-244.

FORMULA

Equals 1199/1200 + 13*Pi/128 - 3*log(2)/4.

EXAMPLE

0.79837428512692106038510479418735875228631658302050...

MATHEMATICA

RealDigits[1199/1200 + 13*Pi/128 - 3*Log[2]/4, 10, 100][[1]]

PROG

(PARI) 1199/1200 + 13*Pi/128 - 3*log(2)/4 \\ Michel Marcus, Oct 29 2021

CROSSREFS

Cf. A093072, A341942.