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%I #10 Jan 12 2026 22:28:46
%S 5,1,3,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,
%T 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,
%U 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8
%N Decimal expansion of 37/72.
%C If three fair six-sided dice are rolled, the probability that the numbers obtained can be the side lengths of a triangle is 37/72.
%C When three dice are rolled, there are 6^3 = 216 triples (a, b, c) with 1 <= a, b, c < = 6. Among these, exactly 111 triples satisfy the triangle inequalities: a + b > c, b + c > a and a + c > b. Hence the probability is 111/216 = 37/72 > 1/2.
%C Among the 111 admissible triples, 42 correspond to scalene triangles, 63 triangles with exactly two equal sides and 6 to equilateral triangles.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F G.f.: (5 - 4*x + 2*x^2 + 5*x^3)/(1 - x). - _Stefano Spezia_, Jan 08 2026
%e 37/72 = 0.513888888...
%t RealDigits[37/72, 10, 100][[1]]
%Y Cf. A021220 (1/216).
%K nonn,cons,easy
%O 0,1
%A _Gonzalo MartÃnez_, Jan 06 2026