We introduce pn-random qn-proportion Bulgarian solitaire (0 < pn, qn ≤ 1), played on n cards distributed in piles. In each pile, a number of cards equal to the proportion qn of the pile size rounded upward to the nearest integer are candidates to be picked. Each candidate card is picked with probability pn, independently of other candidate cards. This generalizes Popov’s random Bulgarian solitaire, in which there is a single candidate card in each pile. Popov showed that a triangular limit shape is obtained for a fixed p as n tends to infinity. Here we let both pn and qn vary with n. We show that under the conditions q2 npnn/log n → ∞ and pnqn → 0 as n → ∞, the pn-random qn-proportion Bulgarian solitaire has an exponential limit shape.
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