Gokigen Naname
Fill every cell with a diagonal — / or \. A circled number counts the diagonals whose tip touches that lattice point, and the diagonals must never close a loop. Click a cell to cycle \ → / → empty; right-click erases. Every finished board is a spanning forest with exactly 2n+1 trees — the loop rule is a counting theorem wearing a puzzle costume.
Acyclicity is arithmetic
Each cell contributes exactly one edge to a graph on the (n+1)×(n+1) lattice points: \ joins the cell's top-left and bottom-right corners, / joins the other two. So a finished board has n² edges on (n+1)² vertices, and a graph with no cycle has exactly V − E connected components: every legal Slant board is a spanning forest with exactly 2n+1 trees, whatever the clues say. The component count is not a degree of freedom. Measured over 1,500 generated boards: 2n+1 trees, zero exceptions.
Two numbers know more than each knows alone
The count rule reads one clue at a time and it saturates early. The duet rule reads two orthogonally adjacent clues as a single constraint over their six shared cells — the classic “adjacent 3s” and “adjacent 1s” tricks fall out of it as special cases, and so does everything else in that family: it simply enumerates the ≤64 joint assignments and keeps what every survivor agrees on.
A redundant rule that still earns its keep
Ranked incrementally, duet looks essential: +10 points of solve rate on 10×10 boards. Ranked by ablation — full ladder minus one rule — it vanishes: the probe behind it catches every board it would have caught. The cycle rule is the opposite: remove it and the full ladder drops 14 points. Redundant is not the same as useless, though: duet finishes boards without guessing that cycle alone cannot, which is exactly what a difficulty grade wants to know. Both rankings are in the stats; they disagree, and the disagreement is the interesting part.