What is a Futoshiki?
Futoshiki (also called Unequal or More or Less) is a logic puzzle in which you fill a grid so that every row and every column holds each number exactly once, guided by greater-than and less-than signs placed between neighbouring squares.
Also known as: Unequal, More or Less.
How do you solve a Futoshiki?
- 01
Follow the chains first
A run of signs all pointing the same way orders every square in it. On a five-wide board a chain of four signs fixes the whole line, and a chain of three leaves its ends near the extremes: the smallest end is one or two, the largest four or five. Trace every chain before writing anything else.
- 02
Rule out the extremes at every sign
A square on the larger side of a sign can never be one, and a square on the smaller side can never be five. A square on the larger side of two signs cannot be one or two. These are quick and remove a surprising number of candidates.
- 03
Apply the row and column rule to the givens
Every number placed rules itself out of its row and column. Combine that with the sign eliminations and pencil the remaining candidates into each empty square; on a five-by-five that is at most four per square, and usually fewer.
- 04
Look for singles
A square with one candidate takes it. A number with only one possible home in a row or column goes there, even if that square still shows other candidates. Each placement is a new elimination in two lines, so return to the signs after every one.
- 05
Squeeze candidates through the signs
A sign constrains candidates, not just final values. If the larger side of a sign can be at most three, the smaller side is at most two; if the smaller side is at least three, the larger is at least four. Push these limits along a chain and the middle squares tighten without being placed directly.
- 06
Finish with pairs and a last check
Two squares in one line sharing the same two candidates lock those numbers out of the rest of the line, and with one more pass of the signs that usually closes a five-by-five. Every puzzle here has exactly one solution, so a stall means a missed elimination, never a square that has to be guessed.
Common Futoshiki mistakes
- Reading a sign backwards. The open end faces the larger number and the point faces the smaller; a sign read the wrong way corrupts every chain it belongs to.
- Treating a sign as “bigger by one”. Greater than means any larger value; a five can face a one across a sign.
- Forgetting the column rule while chasing row chains. The signs draw the eye along rows, but every placement must also be unique in its column, which is where most contradictions hide.
- Guessing to break a stall. A wrong guess is not caught until several moves later, by which time the board is full of dependent errors; a stall means a missed elimination, usually at a sign.
Where does Futoshiki come from?
Futoshiki was devised in Japan by Tamaki Seto in 2001. The name is Japanese for “not equal”, which is the whole of the rule that sets it apart from a plain Latin square: the clues say only that one square is bigger than its neighbour.
It reached a wide audience in Britain in the mid-2000s, when national newspapers looking for a companion to Sudoku began printing it daily, usually as a five-by-five, the size this site uses. Some collections call it Unequal or More or Less.
Is Futoshiki the same as Sudoku?
No, though both require each row and column to hold every number once. Futoshiki has no boxes and no arithmetic, and its clues are inequalities between neighbouring squares rather than given digits.
How long does a 5×5 Futoshiki take?
A five-by-five typically takes two to six minutes, most of it after the chains and extremes have done their work. The daily here is one five-by-five, new at midnight UTC, free with no sign-up, and it plays comfortably on a phone.
Can a Futoshiki have more than one solution?
A badly set one can, if the signs and givens leave two numbers free to swap. Every board on this site is checked to have exactly one solution before it is published.