  Bulgarian solitaire is played on $n$ cards divided into several piles; a move
consists of picking one card from each pile to form a new pile. In a recent
generalization, $\sigma$-Bulgarian solitaire, the number of cards you pick from
a pile is some function $\sigma$ of the pile size, such that you pick
$\sigma(h)\le h$ cards from a pile of size $h$. Here we consider a special
class of such functions. Let us call $\sigma$ well-behaved if $\sigma(1)=1$ and
if both $\sigma(h)$ and $h-\sigma(h)$ are non-decreasing functions of $h$.
Well-behaved $\sigma$-Bulgarian solitaire has a geometric interpretation in
terms of layers at certain levels being picked in each move. It also satisfies
that if a stable configuration of $n$ cards exists it is unique. Moreover, if
piles are sorted in order of decreasing size ($\lambda_1 \ge \lambda_2\ge
\dots$) then a configuration is convex if and only if it is a stable
configuration of some well-behaved $\sigma$-Bulgarian solitaire. If sorted
configurations are represented by Young diagrams and scaled down to have unit
height and unit area, the stable configurations corresponding to an infinite
sequence of well-behaved functions ($\sigma_1, \sigma_2, \dots$) may tend to a
limit shape $\phi$. We show that every convex $\phi$ with certain properties
can arise as the limit shape of some sequence of well-behaved $\sigma_n$. For
the special case when $\sigma_n(h)=\lceil q_n h \rceil$ for $0 < q_n \le 1$,
these limit shapes are triangular (in case $q_n^2 n\rightarrow 0$), or
exponential (in case $q_n^2 n\rightarrow \infty$), or interpolating between
these shapes (in case $q_n^2 n\rightarrow C>0$).
