KenKen

Fill a Latin square so every outlined cage reaches its target number using its printed operation.

What it is

An n×n grid to be filled with 1–n, no repeats in any row or column (a Latin square). The grid is carved into heavily-outlined cages, each labelled with a target and an operation, like “12×” or “3−”. The digits in a cage must combine to the target using that operation; single-cell cages are givens. Digits may repeat within a cage if the Latin rows and columns allow it.

How to play

Start with single-cell cages and cages with unique digit sets (a two-cell “3−” in a 4×4 must be {1,4} or {2,5}…). Combine cage arithmetic with the Latin constraint — a candidate ruled out of a row by arithmetic elsewhere is ruled out of its cage too. Subtraction and division cages may take their digits in either order.

Purpose

The arithmetic-practice puzzle: it was invented as a classroom tool, and it slots between sudoku (pure logic) and kakuro (pure sums) in a mixed book. In the workspace it shares the whole constraint engine with sudoku — the cages are just extra propagators over the same Latin-square variables.

History

Invented in 2004 by Japanese mathematics teacher Tetsuya Miyamoto as KenKen (“cleverness squared”), designed for “the art of teaching without teaching”. Licensed worldwide from 2008, when The Times began publishing it daily. Also circulated as KenDoku, Calcudoku and Mathdoku.

This implementation

• Spec knobs: size (4–9), max_cage, difficulty, subtraction_and_division toggle.
• Generation: answer first — a Latin square is generated, cages are carved as dominoes-and-larger with sizes weighted small, and each cage’s operation is read off its own digits. After uniqueness is proven, operations are re-rolled uniformly per domino (re-proving uniqueness each swap) so − and ÷ actually appear; without that sweep 2 of the 4 operations nearly vanished.
• Guarantees: exactly one solution over the Latin constraints and cage propagators together; every cage checked against its own answer.
• Difficulty: led by the shared technique ladder with a ±1 band nudge from mean cage size. The spec bounds which bands exist — a 4×4 of dominoes is Kids and cannot be otherwise; Hard needs a 9×9.