Futoshiki

A Latin square with inequality signs: fill 1..n once per row and column, and obey every printed <.

What it is

An n×n grid holding each of 1..n exactly once per row and column — the Latin-square rule sudoku shares — with chevrons printed between some neighbouring cells. The chevron points at the smaller value. Very few digits are given; the signs do the work.

How to play

Start where a sign chain forces its ends: in a 5×5, a run a < b < c < d < e along a row pins all five. Elsewhere, bounds reasoning narrows candidates — a cell on the large side of a sign cannot hold 1 — and the Latin-square rule finishes the job.

Why it is in the catalogue

The clearest demonstration of the shared engine’s leverage: futoshiki is a one value per cell puzzle, so rows and columns are AllDifferent (unchanged since sudoku), uniqueness and thinning are count_solutions and reduce_while_unique, and the rating is the technique ladder. The only new code is the inequality propagator — about seventy lines of bounds reasoning.

History

Published by Tamaki Seto in Japan (2001) and popularised internationally by The Guardian from 2006 under the name Futoshiki (“not equal”); also sold as Hutosiki and Unequal.

The implementation’s guarantees

• Uniqueness and no guessing, both proved on the board that ships: the ladder must finish it below Backtrack, and is_unique is re-checked independently of the thinning loop.
• Both kinds of clue are thinned: digits first, then signs, each while the ladder can still finish unaided — a futoshiki printed with every possible sign is a chore, not a puzzle, and the test refuses both extremes.
• The answer is checked against the rules directly (obeys) before any search runs, so a bug in the propagator cannot hide behind one in the generator.
• Signs are drawn as paths, not glyphs: the first render used ∧/∨ text for vertical pairs and the embedded font had neither, so every vertical sign silently vanished. A path cannot go missing.
• Rating basis: technique_ladder — a measurement of the work, not a proxy.