Slant (Gokigen Naname)

Fill every cell with a diagonal so the circled numbers count their touching diagonals — and no diagonals form a loop.

What it is

Every cell of the grid gets exactly one diagonal, / or \. Circled numbers sit on the lattice points between cells: each counts how many of the (up to four) surrounding diagonals touch that point. And the diagonals may never close a loop.

How to play

Corners and edges first: a corner point touches one cell, so its number (0 or

1. decides that cell instantly. A “4” takes all four diagonals into its point; a “0” pushes all four away. The loop rule does the quiet work — the smallest possible loop is a diamond of four diagonals (/\ over \/), and avoiding it forces late cells. Track connected chains: joining two ends of the same chain is exactly what a loop is.

Purpose

The only puzzle here where every cell is filled and the clues live on points rather than cells — a genuinely different geometry that exercises the grid renderer. Its loop rule is implemented as a union-find, the same machinery a maze carver uses, shared conceptually if not in code.

History

Nikoli, published as Gokigen Naname (roughly “in a diagonal mood”). Western publishers standardised on Slant, after Simon Tatham’s popular open-source implementation of it.

This implementation

• Spec knobs: rows, cols, clue_share, difficulty.
• Generation: answer first — a loop-free diagonal filling is built (the Forest union-find with an undo log makes trial placements cheap), point counts are read off it, clues are thinned while unique.
• Guarantees: exactly one filling satisfies the printed numbers; the solver was cross-checked against brute force over every diagonal filling of a 2×3.
• A lesson recorded: the star of four diagonals meeting at a point is a tree, not a loop — the actual minimal cycle is the diamond, an assumption the brute-force sweep corrected. Difficulty bands were re-centred on the measured clue-density spread after by-eye thresholds put every board in Hard and cost 8× in retries.