Loop the loops - Insane

Computer Science Level 5

In the world of pencil puzzles, there are many puzzle types where you have to draw a loop on a lattice grid, including Country Road, Masyu, Pure Loop, Slalom, and Yajilin. In most of these puzzles (and all of the linked above), the loop visits some of the cells, passing through the cells' centers, and may not use a cell more than once (which also means no intersections, no touching itself, etc).

Formally, on a polyomino \(P\), a loop is a sequence of \(n \ge 4\) squares \((a_1, a_2, a_3, \ldots, a_n)\) such that all squares \(a_i\) are in \(P\), \(a_i\) and \(a_{i+1}\) share a side for all valid \(i\), \(a_n\) and \(a_1\) also share a side, and all squares in the loop are distinct. Loops are cyclic (it can start from any square in the loop) and don't have any orientation (reversing the loop doesn't matter), thus \((a_1, a_2, a_3, a_4), (a_2, a_3, a_4, a_1), (a_4, a_3, a_2, a_1)\) all describe the same loop.

Determine the number of loops on a \(4 \times 30\) rectangle.

Too difficult? Go back to the hard difficulty. Want a harder challenge? Try doing 5x30 rectangle; nothing new, though. Want an extreme challenge? Try doing a 10x10 square; I haven't solved it.

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