Futoshiki: The Complete Strategy Guide
Futoshiki is a small, elegant logic puzzle that punches above its weight. The grid is just 5x5, but the combination of Latin-square placement and inequality constraints creates a surprisingly deep solving experience. Every move ripples outward through both the row/column rules and the </> signs scattered between cells.
This guide covers everything from reading the inequalities to the chain analysis that solves harder grids without guessing. By the end you'll see Futoshiki as a tight, fast Sudoku variant rather than a curiosity.
How Futoshiki Works
Fill a 5x5 grid with the digits 1 through 5 so that every row and every column contains each digit exactly once — this is the Latin square constraint. Then satisfy the inequality signs: between some adjacent cells, a < or > sign tells you which cell must hold the smaller value.
That's it. No 3x3 boxes (unlike Sudoku). The inequalities replace the box constraint as the source of extra information.
Reading the inequalities
If you see A < B, then the digit in A is strictly less than the digit in B. They can be 1 less or 4 less, but never equal — the Latin square rule guarantees uniqueness anyway. The sign always points toward the smaller of the two cells.
Reading the Grid Like an Expert
Inequality chains lock corners
A chain like A < B < C < D forces strong placements. If all four cells are in the same row of a 5x5 grid, A must be 1 or 2 (it can't be high because it has three larger neighbors after it).
Cells with all-out signs are 1
A cell with < signs pointing away from it on all sides must hold the smallest possible digit it can be — usually 1.
Cells with all-in signs are 5
The mirror — a cell with all neighbors pointing to it must be the maximum (5 in a 5x5).
Beginner Techniques
Chain bounding
If you see A < B < C in a row, you immediately know: A ≤ 3 (since B and C must be strictly larger), B is between A+1 and 4, and C is between A+2 and 5. Bound every chain you see — even single inequalities give one cell a max and another a min.
The extreme cells
A cell with arrows pointing inward from every adjacent neighbor ("all in") must hold the maximum value of the grid — 5 in standard Futoshiki. The mirror — arrows all pointing out ("all out") — locks the cell at 1. Look for these every time the puzzle opens.
Latin-square scan
For each digit 1-5, scan each row and column. If 4 of the 5 cells in a row are filled, the fifth is forced. Same as Sudoku without the box constraint.
Intermediate Techniques
Pencil-mark candidates
For each empty cell, list which digits 1-5 are still possible after applying row/column constraints. Then prune candidates that violate inequalities. After a few passes, many cells have only one or two candidates left.
The forced order
If a row has the inequality A < B and you know A and B are the only two cells left (the other 3 are filled), you can pick from the two remaining digits and force the smaller to A. Useful when 3 of 5 row cells are placed.
Mutual exclusivity
If two cells share a row or column and you know one is X and the other is Y, swapping them might violate an inequality. Often this resolves which way around they go.
Advanced Techniques
Cross-row chain effects
Inequalities can span rows and columns. A cell's value is constrained by both its row neighbors and its column inequalities. Propagate carefully — placing a digit in one row often triggers constraints two rows over.
Min-max bounding
For any cell A, compute its minimum possible value (1 + number of inequality "smaller-than" cells leading into it) and its maximum (5 - number of "larger-than" cells leading out). Often the min equals the max — instant solve.
The contradiction test
When stuck, pick a cell with two candidates, assume one, and propagate. Reach a contradiction → the other is forced. Use this sparingly.
Common Mistakes to Avoid
- Assuming inequalities mean ±1.
A < Bjust means smaller. A could be 1 and B could be 5. Don't read more into the symbol than it says. - Forgetting it's a 5x5 Latin square. No 3x3 boxes. Don't apply box constraints from Sudoku.
- Skipping the extreme cells. All-in and all-out cells are free moves you should always make first.
- Solving inequalities in isolation. Always combine inequality information with the row and column placements you've made.
Quick Reference
- Goal
- Fill a 5x5 grid with digits 1-5 satisfying the Latin square and all inequalities.
- Latin square
- Each digit appears once per row and once per column.
- Inequality
- Strict (< means strictly less, not less-or-equal).
- All-in cell
- Locked at 5 (maximum).
- All-out cell
- Locked at 1 (minimum).
- Chain
- Sequence A < B < C bounds A high and C low.
- First move
- Find every extreme cell, then bound every chain.
How Futoshiki Compares to Other Number Puzzles
Futoshiki is a Latin-square cousin of Sudoku — same row/column uniqueness, but no boxes, and inequalities instead. Kakuro and Cross Sum introduce arithmetic; Futoshiki keeps things pure with just comparisons.
If you enjoy the inequality logic, you'll probably also like Os and Xs — same flavor of binary deduction.
Frequently Asked Questions
What grid size does Futoshiki use?
Puzzle Page Futoshiki is always 5x5 with digits 1-5.
Are there 3x3 boxes like in Sudoku?
No. Futoshiki uses only the Latin-square constraint plus inequalities. No box uniqueness.
How is Futoshiki different from Sudoku?
Same row/column uniqueness, but Futoshiki replaces the 3x3 box constraint with explicit inequality signs between adjacent cells. Generally shorter and more focused.
How long should a Futoshiki take?
5-12 minutes for an experienced solver. Short enough to be a coffee-break puzzle.
Are all Futoshikis solvable without guessing?
Yes. Each puzzle has a unique solution reachable through pure logic.
Where can I see solved examples?
Every daily Futoshiki is archived on our Futoshiki Answers page, with the complete solved grid.