Futoshiki Puzzle

Futoshiki is a Latin square with less-than and greater-than signs. Every row and column holds each digit from 1 to n once, every sign between neighbouring cells must hold, and some cells may start with a given digit. You can play grids from 4×4 up to 7×7, and every puzzle here has exactly one solution that you can reach by logic, with no guessing.

Rules

An n×n grid takes the digits 1 to n. Each digit appears exactly once in every row and exactly once in every column.

Some neighbouring cells have a less-than or greater-than sign between them. The open side of the sign faces the larger digit. A given digit starts filled in and cannot be changed.

The puzzle is solved when every row and column has each digit once and every sign holds.

Controls

Tap a cell to select it. Its row and column are tinted, and any cell with the same digit is shaded a little darker. Tap a digit on the keypad under the board, or type it. Erase, Backspace, Delete or 0 clears the cell, and the arrow keys move around the grid. Given digits can be selected but not changed.

Notes are small pencil marks for the digits a cell could still be. Turn on Notes with the button or the N key, then each digit you tap or type switches that mark on or off. Shift with a digit adds or removes a note without turning Notes on. When you place a digit, that digit’s notes disappear from the rest of its row and column, in the same undo step.

A repeated digit in a row or column and a filled sign that fails both turn red as soon as they appear. A full row or column that holds each digit once turns soft teal. Press Hint for a digit the rules already force, with the idea behind it. Press Check when you want the board scored against the rules. Undo and Redo step through your moves, and Reset clears the grid back to the givens.

Techniques

Easy grids follow from local facts:

• Edge bounds: the lesser end of a sign cannot be n, and the greater end cannot be 1. When one side is already filled, the open side must sit strictly outside that value.
• Naked singles: after a digit leaves its row and column, a cell with only one candidate left takes that digit.

Medium grids also need longer inequality reasoning:

• Chains: following a run of signs clamps candidates until only one digit remains in a cell.
• Hidden singles: a digit that fits only one open cell in a row or column goes there.

Hard grids add candidate sets and a short try:

• Pairs: two cells that share the same two candidates, or two digits that fit only two cells, clear the extras in that row or column.
• House chains: a cell that is less than (or greater than) two others in the same row or column tightens its range further.
• One-step try: if one of the two candidates in a cell leads to a contradiction, the cell takes the other digit; if both ways force the same digit elsewhere, that digit is placed.

Beginner tips

Start with the signs at the edge of the digit range. A cell with < cannot be the largest digit on the board, and a cell with > cannot be 1. When a neighbour is already filled, the open side has to sit outside that value.

Fill any cell that has only one digit left after the row, column, and nearby signs have spoken. On medium grids, walk a short chain of signs before you look for a hidden single in a row or column.

Use Notes on harder boards so pairs and house chains are easier to see. If you are new to Futoshiki, start with the daily 5×5 easy grid, and move up once local edge bounds and naked singles feel natural.