How to Solve Futoshiki (Less-Than/Greater-Than Puzzle)

Futoshiki is a number puzzle where you fill a grid so every row and column holds each digit once — like Sudoku — while obeying the greater-than and less-than signs between cells. This guide shows the rules in plain English, where to start, how to read the inequality chains, and a free number game to warm up — no app or sign-up.

What Is Futoshiki?

A lot of people land on Futoshiki and have the same first reaction: “I get the Latin-square part, but the inequality signs throw me off.” That is a very reasonable place to start, and it resolves faster than you’d expect once you learn to treat the signs as clues rather than obstacles. The signs are your friends, not the hard part — and in a moment you will see exactly why.

Futoshiki (不等式, futōshiki) is a logic puzzle from Japan. According to Wikipedia’s article on Futoshiki, the name means “inequality” in Japanese, the puzzle was developed by Tamaki Seto in 2001, and it is sometimes spelled hutosiki using an alternative romanization. In English it also goes by the names More or Less and Unequal.

The setup is a square grid, typically 4×4 to 7×7. Some cells start with a given digit; the rest are blank. Between certain pairs of adjacent cells, a greater-than (>) or less-than (<) sign sits in the border. Your job is to fill every blank so that each row and each column contains each digit from 1 to N exactly once, and every inequality sign is satisfied. That third constraint is what makes Futoshiki distinct from a plain Latin-square puzzle.

Pure Latin-square logic is covered in how to get better at Sudoku. If you enjoy layered constraints, KenKen strategies covers operator cages and how to solve Kakuro covers sum-per-region logic. Futoshiki’s constraint is purely directional — no arithmetic beyond reading a sign.

The Rules of Futoshiki (in Plain English)

The official Futoshiki tutorial at futoshiki.com describes the inequality constraints as the puzzle’s core mechanic: each sign tells you which of the two neighboring cells must hold a larger value. There are only two rules to internalize.

Rule 1 — Latin-square constraint. Every digit from 1 to N appears exactly once in each row and exactly once in each column. In a 4×4 grid, every row and column holds 1, 2, 3, and 4. No repeats, no gaps.

Rule 2 — Inequality constraint. Wherever a > or < sign appears between two adjacent cells, the relationship must hold in the final grid. If a sign reads A > B, the digit in cell A must be larger than the digit in cell B. Signs only connect horizontal or vertical neighbors — never diagonal.

A quick example: a sign A > B in a 4×4 grid means A cannot be 1 and B cannot be 4. Add B > C and C is already limited to {1, 2}. Two signs, three cells nearly pinned. Futoshiki is always solvable by logic alone — no guessing. If you feel stuck, a constraint is being overlooked; the sections below show exactly where to look.

Where Do You Start? Forced Cells First

The most common mistake beginners make is starting anywhere and hoping for the best. In Futoshiki, the grid has a natural entry order: go where the constraints are tightest.

Look for given (pre-filled) digits first. Any digit already in the grid removes that number from every other cell in its row and column. Scan all given digits before doing anything else and mark off the eliminations.

Find the endpoint cells of long chains. A chain is a sequence of cells connected by consecutive inequality signs — for example, A < B < C < D. The small end of a three-sign chain in a 4×4 grid must hold 1, because only 1 can be smaller than three other distinct digits from {1, 2, 3, 4}. The large end must hold 4. These endpoint constraints are often the fastest single insight in the whole puzzle.

Apply the extreme-value rule to every sign. Any A > B relationship tells you two things immediately:

• A cannot be 1 (the smallest possible value in the grid).
• B cannot be N (the largest possible value, where N equals the grid size).

Run through every sign in the grid and mark these eliminations before you solve a single row or column. You will often find that two or three signs together force a cell to a single digit — sometimes the entire entry sequence becomes clear from this first pass alone.

Use given digits to anchor sign relationships. A given 2 with “2 > X” in a 4×4 grid confirms X = 1 in a single move. These anchored signs are among the cheapest deductions available.

Reading the Inequality Chains

Once you can spot an individual sign’s implications, the next level is reading chains — sequences of connected signs that force a longer stretch of the grid into a narrow range of candidates.

What a chain looks like. Suppose a row in a 5×5 grid contains C₁ < C₂ < C₃ < C₄ — three consecutive signs linking four cells in strictly increasing order. C₁ can be at most 2 (it needs three larger values above it from the set 1–5). C₄ must be at least 4. If row logic confirms C₁ cannot be 2, it is fixed at 1. If 5 is already placed elsewhere in C₄’s row, C₄ is fixed at 4. A four-cell ascending chain in a 4×4 (digits 1–4) is the extreme case: the only valid arrangement is 1, 2, 3, 4 in order — the entire row solves in one move.

Mixed chains. Not every chain runs in one direction. In A > B < C, cell B is a local minimum — smaller than both neighbors. B cannot be 4 in a 4×4, since both A and C must exceed it. Work each pair separately, then let the results interact. Valley cells and peak cells are worth hunting specifically because they carry double constraints from both neighbors.

Propagation and vertical signs. Every time you place a digit or eliminate a candidate, walk back through every sign touching that cell — placements ripple through two or three signs at once. Signs can also sit between vertically adjacent cells; when a cell is touched by both a horizontal and a vertical sign, it carries double constraints and is worth analyzing early.

Combine Signs with Latin-Square Logic (Elimination)

Futoshiki cannot always be solved by reading signs alone, and it cannot always be solved by row-and-column elimination alone. The two techniques feed each other.

Pencil marking (candidate lists). In any cell, write down every digit still possible given what you know. As you apply sign eliminations and row-column logic, candidates drop out. When a cell has only one candidate remaining, place it.

Standard Latin-square moves. If a digit is possible in only one cell of a row or column, place it — this is a “naked single.” If two cells in a row share the same two candidates and no others, neither candidate can appear elsewhere in that row. Our how to get better at Sudoku guide covers these “naked pair” and “hidden single” patterns in detail; they apply identically in Futoshiki.

The cross-technique cycle.

1. Run through all inequality signs; eliminate candidates using the extreme-value rule and chain logic.
2. Run through all rows and columns; eliminate candidates based on what is already placed.
3. Place any cell reduced to a single candidate.
4. Return to step 1 and repeat with the updated grid.

Most beginner and intermediate puzzles yield entirely to this cycle. Shift emphasis dynamically: signs carry more weight early when the grid is sparse; row-and-column sweeps become faster late when candidate lists are short.

Stuck? Recheck the Extremes and Singles

Every Futoshiki solver hits a wall. Before considering a guess — which is never required in a well-formed puzzle — run this checklist. Feeling stuck always means a constraint is being overlooked, not that the puzzle has no solution.

1. Recheck every sign’s extremes. Confirm you have eliminated impossible extreme values from both cells of each sign. A cell may have narrowed from five candidates to two since you last checked — and one may be the extreme the sign forbids. This check resolves more stuck situations than any other.

2. Find hidden singles in rows and columns. Scan each row and column asking “where can digit 1 go? where can digit 2 go?” rather than “what can go in this cell?” The second framing surfaces placements the first misses.

3. Revisit chain endpoints. With more of the grid filled, endpoint cells that had two candidates may now be forced by row or column logic. A cell with {1, 2} at the small end of a chain resolves as soon as 2 is eliminated elsewhere in its row.

4. Place any cell with one candidate and cascade. After each new placement, restart the checklist from step 1.

Start with 4×4 until sign-reading is automatic, then step up to 5×5. Free daily puzzles at futoshiki.com need no account. The same logic transfers to how to solve Shikaku, logic puzzles for adults, and games like Sudoku.

For a grid-clearing logic puzzle with a bit more structure, Clear Sum lets you delete numbers from a grid — clicking each cell to keep or remove it — until every row and column adds up to its own target. Free, no account.

Frequently Asked Questions

Sources: Wikipedia — Futoshiki · Futoshiki.com — Official Tutorial

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