ABC Path

Place A to Y in a 5×5 grid so every letter touches the next, guided by clues around the edges.

What it is

Twenty-five cells, twenty-five letters, and one rule: consecutive letters are adjacent — orthogonally or diagonally, a king’s move. A’s cell is printed inside the grid; the letters around the edges say which row or column each one occupies.

How to play

From A, B must be one of the eight touching cells and in the line its edge clue names — usually one or two candidates. Each placement narrows the next, and the chain propagates: knowing where a letter must be tells you where its successor cannot be, all the way to Y.

Why it is in the catalogue

Another shared-engine puzzle: AllDifferent over the grid is unchanged since sudoku, uniqueness and thinning are the shared routines, and the new code is the adjacency rule plus a small line-membership constraint.

History

A modern puzzle popularised by Grandgames and Janko’s collection, also published as Letter Path and (with digits) as Number Path.

The implementation’s guarantees

• The adjacency propagator reasons forward, not just locally. The first version asked only the weak question — “can this cell hold v, given some neighbour can hold v ± 1” — and the ladder could not finish a board even with every clue printed. The rule that works is the strong one: a letter’s successor lives in the neighbourhood of wherever the letter could be, so pinning one letter confines the next, and since the letters are a single thread that cascades the whole way along it.
• Boards are over-clued, then thinned. Giving each letter one clue (row or column, by coin) under-determines the path — no board was solvable. With both a row and a column clue per letter the full board solves at Trivial, which is the right place to start reducing from: thin from certainly solvable rather than hope a sparse board happens to work.
• Uniqueness and no guessing are proved on the board that ships, and the test additionally checks the ladder’s solution is the generated path.
• Rating basis: surviving_clue_count — named honestly, because one rule means the ladder cannot separate boards.