  The multistochastic $ (n,k)$-Monge--Kantorovich problem on a product space
$\prod_{i=1}^n X_i$ is an extension of the classical Monge--Kantorovich
problem. This problem is considered on the space of measures with fixed
projections onto $X_{i_1} \times \ldots \times X_{i_k}$ for all $k$-tuples
$\{i_1, \ldots, i_k\} \subset \{1, \ldots, n\}$ for a given $1 \le k < n$. In
our paper we study well-posedness of the primal and the corresponding dual
problem. Our central result describes a solution $\pi$ to the following
important model case: $n=3, k=2, X_i = [0,1]$, the cost function $c(x,y,z) =
xyz$, and the corresponding two--dimensional projections are Lebesgue measures
on $[0,1]^2$. We prove, in particular, that the mapping $(x,y) \to x \oplus y$,
where $\oplus$ is the bitwise addition (xor- or Nim-addition) on $[0,1] \cong
\mathbb{Z}_2^{\infty}$, is the corresponding optimal transportation. In
particular, the support of $\pi$ is the Sierpi\'nski tetrahedron. In addition,
we describe a solution to the corresponding dual problem.
