  A formal n-square is the set of positions in an square matrix of size n. A
shuffle of a formal n-square consists of independent rotations of each row and
of each column. A key result turns out to be valid at least for n <= 34 and n =
37: Each set of n positions can be mapped with one shuffle onto a transversal
of the columns. We consider two applications to equi-n-squares (i.e.,
n-matrices filled with digits 0, .., n - 1 in equal amounts).
  First, a shuffled equi-n-square can be seen as a torus with n colors and two
orthogonal layers of n rings that can be rotated. Unlike Rubik's cube, each
permutation of colored cells can be implemented with shuffles. An upper bound
of $3*(-1)^{n-1} + 6n$ shuffles is derived from the key result.
  Our second application invokes column transversals and a process of
indirection to produce theoretically unpredictable sequences of integers in
shuffled equi-n-squares.
  Our proof of the key result involves optimizing position sets, averaging,
computations based on number partitions, rotating subsets of a regular $n$-gon
apart, and the use of cyclotomic polynomials. A few intermediate results need
computer assistence. These efforts also generated a variety of (partially)
unsolved problems. We selected eight of these for a brief discussion based on
the available theoretical and computer evidence.
