Futoshiki 不等式

Futoshiki (不等式, the Japanese word for inequality) is a Latin square puzzle. You fill an N×N grid with the numbers 1 to N so that each row and each column contains every number exactly once, and every greater-than or less-than sign printed between two neighbouring cells must be true. A handful of numbers are given at the start; the signs do the rest of the work.

Futoshiki emerged in Japan around 2001. It reached Britain in September 2006, when The Guardian began printing it beside Sudoku after Tamaki Seto, Japanese editor at the puzzle publisher Kandour, brought it over. It is also known as More or Less, Unequal or Hutoshiki.

On LogicPuzzles.io you get one daily 6×6 Futoshiki that is the same for every player, plus unlimited 5×5, 6×6 and 7×7 grids. Every puzzle in our banks is checked by two independent solvers to have exactly one solution, and by a deduction grader to be solvable without guessing. Pencil marks, undo, a timer and a Check button are built in, and your progress is saved in this browser.

Start with the signs rather than the givens. A cell on the small side of a sign can never hold N, and a cell on the large side can never hold 1. Chains of signs are stronger still: three cells linked as a < b < c force a to be at most N minus 2 and c to be at least 3, and the longer the chain, the tighter the squeeze. A chain of N cells, running the full width of the grid, is fixed outright, and a chain of N minus 1 cells leaves each cell exactly two candidates.

Once the signs have trimmed the candidates, treat the grid like a small Sudoku. Look for a row or column in which a number can go in only one cell (a hidden single), and for a cell that has only one candidate left (a naked single). Pencil marks help here: switch on Pencil or press P, tap the candidates you want to keep, and watch the keypad, which dims a number as soon as it has been placed N times.

The most common mistake is reading a sign the wrong way. The open side of the symbol always faces the larger number, so 3 > 1 and 1 < 3 mean the same thing whichever way the grid is drawn. The Check button highlights any sign that is currently broken and any duplicate in a row or column, and it also tells you whether the numbers you have entered so far match the solution.

Futoshiki emerged in Japan around 2001. The name is the Japanese word for inequality, a reference to the comparison signs that replace most of the givens of an ordinary Latin square puzzle. Tamaki Seto, Japanese editor at the puzzle publisher Kandour, brought it to Britain, where The Guardian began printing it in September 2006 during the Sudoku boom. It remains a regular in British puzzle pages: The Times, The Daily Telegraph and the i newspaper also run it. It belongs to the same family as Sudoku and KenKen: all three are Latin squares with an extra constraint, and techniques such as naked singles and hidden singles transfer directly between them.