It appears that a variation on [Bulgarian solitare](http://en.wikipedia.org/wiki/Bulgarian_solitaire) has a fixed point regardless
of the starting $n$.

For example, let $n=69$, and consider this partition:
$$
(8,8,7,7,5,5,5,5,5,4,3,3,2,2)
$$
In Bulgarian solitare, $1$ would be removed from each "stack/pile" to form another stack.
In the variation, I remove $1$ just from the $k=9=3^2$ largest stacks, $(8,8,7,7,5,5,5,5,5)$. That reduces those stacks to $(7,7,6,6,4,4,4,4,4)$ and
adds a stack of size $9$.

I find it easier to visualize this process with the stacks organized in an array with the largest
stacks surrounding the upperleft corner. Then $1$ is removed from
the $3 \times 3$ square of stacks including the upperleft corner.
The numbers are resorted and the same process applied again,
always gathering from the largest $k=9$ stacks:
$$
\left(
\begin{array}{cccc}
 8 & 8 & 5 & 4 \\
 7 & 7 & 5 & 3 \\
 5 & 5 & 5 & 3 \\
 0 & 0 & 2 & 2 \\
\end{array}
\right)
\;\rightarrow\;
\left(
\begin{array}{cccc}
 9 & 7 & 6 & 4 \\
 6 & 7 & 4 & 4 \\
 4 & 4 & 4 & 3 \\
 0 & 2 & 2 & 3 \\
\end{array}
\right)
\;\rightarrow\;
\left(
\begin{array}{cccc}
 9 & 8 & 5 & 3 \\
 6 & 6 & 5 & 3 \\
 3 & 4 & 4 & 3 \\
 2 & 2 & 3 & 3 \\
\end{array}
\right)
\;\rightarrow\; \cdots
$$
The endpoint of this process, after $20$ steps, is:
$$
\left(
\begin{array}{cccccc}
 9 & 8 & 5 & 1 & 1 & 1 \\
 6 & 7 & 4 & 1 & 1 & 1 \\
 1 & 2 & 3 & 1 & 1 & 1 \\
 1 & 1 & 1 & 1 & 1 & 1 \\
 1 & 1 & 1 & 1 & 1 & 1 \\
 0 & 0 & 0 & 1 & 1 & 1 \\
\end{array}
\right)
$$
This seems to be a fixed point regardless of $n$, as long as the list of
stacks is longer than the largest $k$ being reduced at each step. 
(If $k$ encompasses
the entire list of stacks, this just reduces to Bulgarian solitare, and cycles
rather than fixed points can occur.)

My question is: Is this true, that the process described
leads to the fixed point
$$(k, k{-}1, k{-}2, \ldots, 3,2,1,1,1,\ldots,1)$$
under those conditions?
It's a bit surprising to me that it doesn't lead to cycles for non-triangular $n$.
Perhaps it does? I have not explored extensively
(and I've only looked at $k$ which are squares).