> 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.
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.
- 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 922One can think of the set of all possible board configurations as the vertices as a graph, with edges indicating how to move between configurations. Then your 1536 solutions are the 1536 distinct shortest paths between the starting and target configuration.
Then, you can also choose to consider not just board configurations, but board configurations up to simultaneous permutation of rows and columns; that will also reduce the number of unique solutions.
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?
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?
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?
Alternatively, you could use 7 col. Your 4 row ops are equivalent to ('col', 3, 0), ('col', 0, 2), ('col', 2, 1), ('col', 1, 0).
Feature request: I was expecting an animation (three stars and confeti!) or at least a congratulation message when I won.
For the 4x4 board there is only 2^16 nodes, and 8*2^16 edges, so you can materialize the graph and get away with brute-forcing the whole graph.
But for bigger boards you won't be able to materialize the whole graph.
Maybe there are better heuristics to be found than the simple hamming distance. You should try have an AI look for them, by comparing the performance of a RL path planning vs the basic heuristic.
...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.