Futoshiki Rules: How to Play, Strategies and Variations

What is Futoshiki?

Futoshiki is a Japanese number puzzle played on a small square grid, most often 5×5. You fill every cell with a digit so that each row and each column contains every number exactly once, much as in Sudoku. The twist is a set of inequality signs printed between some pairs of neighboring cells. Each sign tells you which of the two cells holds the bigger number.

The name comes from the Japanese 不等式 (futōshiki), which means "inequality". English-language publishers sometimes print it as More or Less. Because a finished grid uses each digit once per row and column, a solved Futoshiki is a Latin square, the same mathematical structure that sits underneath Sudoku and KenKen.

What makes Futoshiki special is the kind of thinking it asks for. Sudoku is mostly about where a number can go. Futoshiki is about order: which of two cells is larger, and how far a chain of signs pushes a value up or down. That relational logic gives even a 25-cell grid real depth, and it is why many solvers treat Futoshiki as the perfect short puzzle for a coffee break.

A short history

Futoshiki is a young puzzle. It was developed in Japan by Tamaki Seto in 2001, according to the Wikipedia article on Futoshiki, and it spread through puzzle magazines before crossing to Europe.

It found an especially warm home in the United Kingdom, where it runs in several national newspapers, including The Times, The Guardian and The Daily Telegraph. The appeal to editors is easy to see: the rules fit in two sentences, the grid takes up little space on the page, and the difficulty can be tuned simply by adding or removing signs and given digits.

Futoshiki belongs to a family of Latin square puzzles that share the same backbone of one digit per row and column. Sudoku adds 3×3 boxes, KenKen adds arithmetic cages, and Futoshiki adds inequalities. If you already know one of them, you already know half of the others.

The rules of Futoshiki

Futoshiki has only a handful of rules. For a grid of size N (N is 5 in the standard puzzle), they are:

1. 1Use the numbers 1 to NEvery cell gets exactly one digit from 1 to N. On a 5×5 grid that means 1, 2, 3, 4 and 5, with no zeros and nothing larger.
2. 2No repeats in a rowEach row contains every number from 1 to N exactly once.
3. 3No repeats in a columnEach column also contains every number from 1 to N exactly once. There are no boxes or regions to worry about, only rows and columns.
4. 4Obey every signAn inequality sign between two neighboring cells must be true. The sign always points at the smaller number, with its open side facing the larger one. Vertical signs work the same way: a sign pointing down means the upper cell is larger.
5. 5Givens stay fixedAny digit printed in the starting grid is part of the solution and cannot be changed. Some puzzles have no givens at all and rely on the signs alone.

Where two neighbors have no sign between them, there is no relationship to respect: either one may be larger. A well-made Futoshiki has exactly one solution, and you can always reach it by logic without guessing.

A worked example

Here is a complete 5×5 puzzle with only two given digits and eight signs, next to its unique solution. Try it on paper before you study the right-hand grid.

A good place to start is the second row. Its chain of two signs says the first cell is smaller than the second, which is smaller than the third. The sign below the first cell adds that it is larger than the cell underneath, so it cannot be 1. It must be at least 2, which pushes the middle cell of the chain to at least 3 and the third cell to at least 4. Meanwhile, in the bottom row the givens 5 and 1 leave only 2, 3 and 4 for the other three cells, and the sign on the left says the first of those is smaller than the second. Small squeezes like these, repeated across the grid, are how every Futoshiki is solved.

How to play: a step-by-step method

There is no single correct order, but this routine works on almost every Futoshiki you will meet:

1. Pencil in candidates. Write the possible digits small in each empty cell, or at least in the cells you are working on. Futoshiki is far easier with notes than in your head.
2. Clear the givens from their lines. A given 4 removes 4 from every other cell in its row and its column.
3. Apply every sign. The larger side of a sign can never be 1, and the smaller side can never be N. Cross those candidates out everywhere a sign appears.
4. Follow the chains. When several signs line up in the same direction, push the limits along the chain. In a three-cell chain on a 5×5 grid, the largest cell is at least 3 and the smallest is at most 3.
5. Look for singles. If a cell has only one candidate left, or a digit has only one possible home in a row or column, write it in.
6. Recheck the neighbors after each placement. Every new digit tightens the cells around it. A 2 placed on the larger side of a sign instantly forces a 1 on the other side.

Strategies strong solvers use

Once the basic routine feels natural, these techniques cover nearly everything you will meet in published puzzles.

Most puzzles marked easy or medium never need the last technique. If you find yourself testing values often, go back and look for a chain or a pair you have missed.

Popular variations

The core rules are easy to extend, and you will find several versions in newspapers, books and apps:

• Different grid sizes. Beginner puzzles use 4×4 grids, the standard is 5×5, and harder puzzles run to 6×6, 7×7 or larger. Each extra row adds a digit, and chains leave more room for doubt, so difficulty climbs quickly with size.
• Signs only. Some Futoshiki start with no given digits at all. The inequalities alone determine the unique solution, which makes chain reasoning essential from the first move.
• Greater Than Sudoku. A 9×9 Sudoku that keeps the 3×3 boxes but replaces the given digits with inequality signs between neighboring cells inside each box. It combines both puzzles' logic and is a natural next step once 5×5 Futoshiki feels easy.
• Arithmetic cousins. KenKen keeps the Latin square but swaps the signs for cages that must reach a target by addition, subtraction, multiplication or division. Kakuro drops the row-and-column rule entirely and uses sum clues in a crossword-shaped grid.

Common mistakes to avoid

Most Futoshiki errors come from a few habits, and all of them are easy to fix:

• Reading a sign backwards. The point always faces the smaller number. If in doubt, read the pair aloud from left to right or top to bottom: "four is greater than two".
• Ignoring the vertical signs. Horizontal signs are easier to spot, so solvers often forget the ones between rows. Scan both directions every time you update a cell.
• Reading meaning into a missing sign. No sign means no relationship. It does not tell you which cell is larger.
• Looking at signs one at a time. A single sign rules out very little. A chain of three or four signs can pin down several cells at once, so always trace the full chain.
• Forgetting the Latin square. The signs get all the attention, but many placements come from plain row and column uniqueness. Check for singles after every move.

Futoshiki compared with Sudoku and KenKen

All three puzzles rest on the rule of one digit per row and column, so moving between them is easy. Sudoku uses a fixed 9×9 grid with 3×3 boxes and relies on given digits. KenKen uses cages that must hit a target number with arithmetic. Futoshiki needs no arithmetic at all, only comparisons, and its grids are small enough to finish in a few minutes.

That makes it a good bridge for solvers who enjoy Sudoku's scanning but want a more visual, relational kind of deduction. It also suits anyone who finds arithmetic cages tiring. If the ordering logic hooks you, Kakuro is a good next step for sum-based reasoning, and Star Battle offers pure placement logic without any numbers.

Difficulty in Futoshiki depends mostly on grid size and how many clues remain. An easy 5×5 with plenty of signs takes a few minutes, while a sparse 7×7 can hold your attention far longer. Start small, get comfortable with chains, then move up a size.

Frequently asked questions