...Some time later... This is quite hard!

I think thinking about this puzzle as Gaussian elimination is not helpful!

I think the controls would work better if you dragged the row/column onto the one want to change.

However I'm sure there is a diverting puzzle game in here somewhere. I wonder if you used narrative language and symbolism unrelated to linear reversible circuit synthesis (but kept whatever mechanic is important) an average player might be able to grasp it more easily?

A (non-optimal, but straightforward) procedure for doing so is like so: First, use Gaussian elimination row-wise to put any matrix into reduced row echelon form. One can now use Gaussian elimination column-wise to transform the matrix into a 2x2 block matrix whose upper-left block is an identity matrix (of size corresponding to the rank) and whose other blocks are zero. Since all moves are invertible, any two matrices of the same rank are thus connected via the same such block matrix.

In general, it is necessary to use both row and column moves. However, if both matrices are square with full rank (as in today's puzzle), one can just use row moves (or just as well, just use column moves), using just Gaussian elimination. More generally, one can just use row moves iff both matrices have the same row space, and similarly for columns.

Feature request: I was expecting an animation (three stars and confeti!) or at least a congratulation message when I won.

With 8 moves and rows only: 2->1, 1->2, 2->1, 3->2, 2->3, 4->3, 4->1, 1->4.

A more efficient solution should be possible; did anyone find any?

> The game is inspired by the synthesis of linear reversible circuits; a problem in reversible and quantum computation. Here, the goal is to construct a target operation, the target pattern in Swapple, using a sequence of simpler operations, specifically controlled NOT (CNOT) gates, which flip the state of a target bit if and only if a control bit is set. In Swapple, each row and column operation corresponds to applying a CNOT gate. Your task is to find a sequence of these gates, i.e. a circuit, that transform the initial configuration, corresponding to an empty circuit, into the target configuration. Moreover, finding one of the shortest sequences of moves to achieve this goal corresponds to finding one of the most efficient circuits that implements the desired operation.

- there are 1536 solutions

- almost all moves are useful, non are required

- for every row-xoring move there is exactly one column-xoring move that appears in the same number of solutions (and no move appears twice in a solution)

Here is the number of solutions a move appears in (0-based indices):

  C3→2    R2→3    0
  C3→1    R2→1    82
  C2→0    R3→0    93
  C0→3    R0→2    163
  C2→1    R3→1    342
  C1→3    R1→2    426
  C1→2    R1→3    558
  C3→0    R2→0    614
  C2→3    R3→2    640
  C1→0    R1→0    726
  C0→1    R0→1    810
  C0→2    R0→3    922

Did you make some assumptions about the minimum window / screen size based on oversized modern smartphones, forgetting that lots of us still cling to more reasonably sized older devices?