Free Printable Futoshiki Puzzles PDF

The Mathematics and Strategy of Futoshiki

Futoshiki (不等式, literally translated from Japanese as 'inequality') is a discrete combinatorial puzzle that combines the row-and-column Latin square constraints of Sudoku with strict arithmetic order relations. Originating in Japan during the puzzle renaissance of the early 2000s, Futoshiki challenges solvers to complete an N×N board using digits 1 through N such that each integer appears exactly once in each row and column, while obeying all inequality signs placed between adjacent cells.

1. The Latin Square Rule

Every 6×6 Futoshiki board is governed by the Latin square property. In an N=6 grid, every row and column must contain the integers [1, 2, 3, 4, 5, 6] with zero omissions and zero duplications. Any digit placed in a cell immediately eliminates that value from the other five cells in the same horizontal row and vertical column.

2. Transitive Inequality Chains

The defining characteristic of Futoshiki is the presence of greater-than (>) and less-than (<) operators between adjacent cells. Because inequalities are strictly transitive, a sequence of interconnected signs forms an inequality chain: A > B > C > D.

In a 6×6 puzzle, where candidate values span from 1 to 6:

• A 4-cell chain (A > B > C > D) immediately eliminates 1, 2, and 3 as candidates for cell A, bounding it strictly to (4, 5, 6).
• Simultaneously, cell D cannot hold 4, 5, or 6, bounding it strictly to (1, 2, 3).
• If an inequality chain stretches across 5 cells, the endpoint values are uniquely fixed or bounded to narrow binary choices before any row-column eliminations occur.

3. The Extremum Elimination Technique

Before writing any numbers, professional solvers conduct an initial scan for extreme values (1 and 6):

• The 'No 1' Principle: Any cell located on the larger side of an inequality symbol (cell > neighbor) can never be 1, because no valid positive integer in the puzzle domain is strictly smaller than 1.
• The 'No 6' Principle: Any cell located on the smaller side of an inequality sign (cell < neighbor) can never be 6, because no integer in the 6×6 grid is strictly larger than 6.
• Convergent Arrows: A cell that receives multiple incoming 'less-than' signs from different directions is forced toward the lowest remaining available numbers in that row.

4. Row-Column Intersections and Naked Pairs

Once candidate lists are established, apply classical constraint satisfaction methods. If two cells in the same row share identical candidate pairs (such as [2, 5]), those two numbers are locked into those positions and can be eliminated from all other cells in that row. Cross-referencing these naked pairs against inequality pointers frequently forces naked singles in adjacent columns.