Futoshiki

Futoshiki is a Latin-square logic puzzle: on an n×n grid you fill every cell with a number from 1 to n so each number appears exactly once in every row and once in every column, while also obeying the greater-than and less-than signs printed between some cells. Those two rules are the whole game; it is also published as Unequal, and sometimes transliterated Hutoshiki.

What is the Latin-square rule?

The first rule is the “Latin square”: no number may repeat along any row or column, the same idea that underpins Sudoku and Skyscrapers. On a 4×4 board that means each row and column holds 1, 2, 3 and 4 exactly once; on a 5×5, the numbers 1 to 5.

How do the inequality signs work?

What sets Futoshiki apart is its second rule. Between some pairs of neighbouring cells you will find a greater-than or less-than sign — an inequality. Wherever one of these signs sits, the two cells it joins must always obey it: the value on the open, wide side of the sign is the larger number, and the value on the pointed side is the smaller one. These inequalities never go away — they have to hold true in the finished grid, not just at the start.

Why can every board be solved by logic?

To give you a foothold, a few cells may already hold given numbers when the puzzle begins. Everything else you work out yourself. The two rules pull against each other in a productive way: the Latin-square rule limits which numbers can live in a row or column, and the inequality signs decide the order they must fall in. Because those constraints are precise and a well-made board is built to have exactly one solution, every Futoshiki can be cracked by pure reasoning — there is always a provable next step, and you never need to guess. Every board on this page is generated and checked to guarantee that single, logical answer.

The sign always opens its wide mouth toward the larger number and points its narrow tip at the smaller one: 3 > 1 reads “three is greater than one”, and 1 < 3 reads “one is less than three”, whichever way the sign faces between two adjacent cells. The whole puzzle turns on reading these correctly, so it is worth slowing down on them. The greater-than and less-than signs are the same symbols you meet in maths: > means “greater than” and < means “less than”. The simplest way to remember them is by shape — the big end always eats the bigger value.

Which cell holds the smaller value?

On the grid these signs sit between two adjacent cells, either horizontally or vertically, and the same rule applies whichever way they face. If a sign points at a cell, that cell holds the smaller of the pair; if a cell sits on the open side, it holds the larger. From this you get two immediately useful facts. A cell on the small side of a sign can never be the maximum value n — something has to be bigger than it. And a cell on the large side can never be 1 — something has to be smaller than it.

How do chains of signs help?

Signs also link up into chains. If you read a < b < c across three cells, then a is smaller than b and b is smaller than c, so the three values must rise steadily. In a 4×4 grid a chain of three such cells almost fixes itself: the lowest in the chain is squeezed toward 1 or 2 and the highest toward 3 or 4. Learning to read these greater-than and less-than signs at a glance, and to picture the order they force, is the single most valuable habit in Futoshiki.

Solve Futoshiki by combining the inequalities with plain Latin-square elimination: let inequality chains fix the extremes, work out which numbers a row or column can take and then use the signs to order them, exploit signs at the edge of a line, anchor on given numbers, and always make every forced move before looking harder. A handful of techniques will carry you through almost any board:

• Let inequality chains fix the extremes. A run like a < b < c needs at least three distinct rising values, so the smallest end is pushed toward 1 and the largest end toward the top number. In a 4×4 a chain of three cells often pins both ends straight away.
• Combine signs with Latin-square elimination. Work out which numbers a row or column can still take, then use the signs to decide their order. The two rules together frequently leave only one legal arrangement.
• Watch a sign at the end of a row. A > or < sitting at the very edge of a row or column limits the candidates hard, because the cell beyond it has nowhere to hide a smaller or larger partner — an edge cell on the small side often must be 1, and on the large side often must be n.
• Use given numbers as anchors. Any starting digit removes itself from the rest of its row and column at once, and it also fixes the value on the other side of any sign attached to it. Build outward from those anchors.
• Count from both ends. If a cell cannot be the maximum (it is on the small side of a sign) and also cannot be 1 (a given elsewhere blocks it), its range narrows from both directions until one value remains.

In what order should you make moves?

The winning habit is the same as in every logic puzzle here: make every forced move first — the cells you can prove — and only then look harder at the rest. Never guess. On a properly made Futoshiki there is always a certain next step waiting; if you feel stuck, re-read the signs around the emptiest row and the answer usually surfaces.

A short walk-through shows the two rules feeding each other: take the top row of a 4×4 board holding 1, 2, 3 and 4, with the first three cells linked by signs reading a < b < c, and watch the chain plus column elimination assemble the grid with no guessing. The values rise from left to right across those three cells.

How does the chain pin the ends?

Start with the chain. Three cells that climb steadily need three different numbers in increasing order, so the leftmost cell a cannot be 3 or 4 (there must be room for two larger values above it), and the third cell c cannot be 1 or 2 (there must be room for two smaller values below it). That already squeezes a toward 1 and c toward 4. If a given number or a column clash rules 1 out of the fourth cell, then a must be 1, which forces c to be 4 and leaves b as 2 or 3.

How does column elimination finish it?

Now bring in Latin-square elimination. The fourth cell in the row is whatever is left after a, b and c, so once the chain settles it is decided too. Drop down a column: each value you have just placed removes itself from every other cell in its column, which often forces a number there, which in turn feeds back into the next row. That back-and-forth — read the signs to fix an order, then eliminate down the rows and columns — is the entire rhythm of Futoshiki, and it assembles the grid without a single guess.

This page offers Futoshiki at two sizes — 4×4 (numbers 1–4) and 5×5 (numbers 1–5) — and within a size the real difficulty lever is how many given numbers and signs the board hands you: more help solves briskly, fewer givens and fewer inequalities play much tougher. Both fit comfortably on a phone screen, and each tap simply cycles a cell through its values, so the bigger board is no harder to handle — only deeper to reason about.

• 4×4 uses the numbers 1–4 and is the gentler board, perfect for getting used to reading the signs. Short inequality chains often pin both of their ends quickly, so solutions come together at a satisfying pace.
• 5×5 uses 1–5 and gives the logic more room to breathe. There are more cells to track and longer chains to follow, so the deductions stretch further before everything locks into place.

What makes one board harder than another?

Within a size, the real difficulty lever is how much help the board hands you. A puzzle with several given numbers and plenty of signs offers strong footholds and solves briskly; a sparser board — fewer givens and fewer inequalities — forces you to lean harder on chains and elimination, and plays much tougher. Tap New for a fresh board at either size whenever you like, use Reset to clear your entries and start the same board again, or play the shared Daily to take on exactly the same puzzle as everyone else that day.

Futoshiki, Unequal and Hutoshiki are all the same puzzle — a Latin square with inequality signs — so any tip you pick up under one name works under the others. Its Japanese title, Futoshiki, means roughly “not equal” or “inequality”, which describes the puzzle perfectly; you will also see it transliterated Hutoshiki, and in the English-speaking world it is widely published as Unequal.

Futoshiki sits firmly in the Latin-square family, which makes it close kin to several puzzles in this collection. If you enjoy filling a grid so each value appears once per row and column, try Sudoku, which keeps the no-repeat rule and adds 3×3 boxes, or Skyscrapers, which keeps the same grid rule but replaces the inequality signs with clues about how many towers you can see from each side. The deductive habits carry straight across; Futoshiki simply expresses its extra constraint as a row of greater-than and less-than signs.

The core Futoshiki vocabulary is short: a Latin square is a grid where each value appears once per row and column, an inequality is a greater-than or less-than sign fixing which of two cells is larger, a given is a number printed at the start, and a candidate is a value a cell could still legally take. A few words turn up in almost every Futoshiki guide:

• Latin square — a grid in which each value appears exactly once in every row and every column. It is the backbone of Futoshiki and of its number-placement relatives.
• Inequality (constraint) — a greater-than or less-than sign between two adjacent cells that fixes which of the pair is larger. Each sign is a hard constraint that must hold in the finished grid.
• Given — a number printed in a cell at the start. Givens are fixed clues you build outward from; they cannot be changed.
• Candidate — a value a cell could still legally take. Solving is largely the act of whittling each cell’s candidates down — using the signs and the no-repeat rule — until just one remains.

You do not need the vocabulary to play — just tap a cell to cycle its value — but it makes strategy guides much easier to follow.

Most Futoshiki trouble comes from a few avoidable habits — misreading a sign, obeying a sign but repeating a number in the row, forgetting the extremes, guessing, and ignoring the chains — and naming each one removes the error before it costs you. They are easy to break once you can spot them:

• Misreading a sign. The classic slip is forgetting which way the symbol points. Remember the wide side opens toward the larger number and the tip points at the smaller — read every < and > deliberately.
• Obeying a sign but breaking the row. It is easy to satisfy an inequality and accidentally repeat a number elsewhere in that row or column. Both rules must hold at once; check the line as well as the sign.
• Forgetting the extremes. A cell on the small side of a sign can never be the maximum value, and a cell on the large side can never be 1. Skipping these free deductions leaves easy progress on the table.
• Guessing. Placing a number you cannot justify can carry you a long way before it clashes. There is always a forced move — find it instead.
• Ignoring the chains. Treating each sign in isolation misses the power of a run like a < b < c, which can pin several cells at once.

Futoshiki is a small puzzle with a surprising amount of depth: it blends the clean satisfaction of completing a Latin square with the extra spark of ordering, has no arithmetic and no luck, and gives every board exactly one solution reachable by reason — a calming, screen-friendly few minutes that also works the mind. Every sign is a tiny relationship to reason about, and clicking the last number into place feels genuinely earned.

Is Futoshiki good mental exercise?

It is also a real workout for the mind. Tracking candidates, combining the inequalities with row-and-column elimination, and thinking a step or two ahead all sharpen the same logical muscles you use far from the grid. Because every board has exactly one solution reachable by reason, progress always feels fair. A 4×4 is a brisk five-minute reset; a 5×5 is a more absorbing solve; and the shared Daily gives you the same board as everyone else and a small streak to keep. If you take to it, the same no-repeat instinct carries straight over to Sudoku and to the visibility twist of Skyscrapers.

Futoshiki is a relatively modern Japanese puzzle, created by the designer Tamaki Seto, whose name means roughly “not equal” or “inequality” in Japanese — also transliterated Hutoshiki — and it reached English-speaking solvers as Unequal, appearing in UK newspapers including The Guardian from around 2006. The name is a neat description of the greater-than and less-than signs at its heart, which is why you will occasionally see the puzzle published as Hutoshiki.

In UK newspapers — among them The Guardian — it joined Sudoku and the other grid puzzles that had made the Latin square a daily fixture from around 2006. What has never changed is the elegant core that made it popular in the first place: fill a grid so each number appears once per row and column, obey a scattering of inequality signs, and reach the single answer by pure logic.