How to play Futoshiki

Futoshiki is a Japanese logic puzzle whose name means "not equal". You fill a square grid so that every row and every column contains each digit exactly once - and between some neighbouring cells sits a sign that says which of the two digits is larger. A few digits may be given at the start; the signs do most of the work.

Every puzzle here is generated fresh by the game when you start it, has exactly one solution, and can be solved by reasoning alone. You never need to guess. The four levels grow from a 5x5 grid on Easy to 7x7 grids on Hard and Crazy, and each level asks for deeper reasoning than the one before.

How to play on skill.games

When you open the game, the settings popup appears. Choose a level under DIFFICULTY, choose how mistakes are treated under MISTAKES, and press PLAY. A new puzzle is generated in a moment.

• The board takes up most of the screen. The cells stand slightly apart, and the signs sit in the gaps between them. A sign always points at the smaller digit, like a "less than" or "greater than" sign: its open side faces the larger one. Signs between rows point up or down. Given digits are printed in dark ink and can't be changed; the digits you enter appear in blue.
• Choosing a cell: tap any cell. Its row and column are shaded, and every cell holding the same digit is highlighted.
• Entering a digit: with a cell chosen, tap a number on the number pad (under the board, or beside it when your screen is wider than it is tall). The small number under each key shows how many of that digit are still to place; a key fades out once every row has it.
• NOTES: tap NOTES to switch notes on. Numbers you tap are now pencilled into the cell as small candidates, and tapping one again removes it. Tap NOTES again to go back to entering digits. When you enter a digit, it is removed automatically from the notes of every cell in its row and column.
• ERASE clears the chosen cell (its digit or its notes).
• UNDO takes back your last action - a digit, a note or an erase - all the way back to the start. Undo is free and unlimited.
• Highlighting a digit: with no cell chosen, tap a number on the pad to highlight every cell and note holding that digit. Tap it again to clear the highlight.

On a computer you can also use the keyboard: the arrow keys (or W, A, S, D) move between cells, the number keys enter a digit, Shift and a number pencil a note, Backspace, Delete or 0 erase, N switches notes on and off, and Z undoes.

The top line shows the clock, the level, the grid size and your mistakes. The clock runs only while you play: it stops when the menu or a popup is open. The menu button in the top-left corner pauses the game, starts a new puzzle or switches the sound on or off. The game is saved after every move, so if you close the page you can press CONTINUE later and carry on with the same grid, notes, clock and undo history.

When the last cell is filled correctly, the board ripples and the results screen shows your time, your best time at that level (a star marks a new record), your mistakes, the grid size and the puzzle's code, together with your wins, losses, win rate and streaks.

Rules

The grid. On a grid of size N (5 to 7 here) every row and every column must end up holding each digit from 1 to N exactly once.

Signs. Where a sign stands between two cells, the digits in them must obey it: the cell the sign points at holds the smaller digit. Cells with no sign between them can hold any digits the rows and columns allow.

Givens. Some puzzles start with a few digits already placed. They are part of the puzzle and can't be changed. Every puzzle has exactly one way to complete it.

Mistakes - Unlimited (the default). If you enter a digit that is already in the same row or column, it turns red and the mistakes counter goes up by one. If a digit you enter breaks a sign with the digit next to it, the sign turns red on a pale red disc and that also counts as a mistake. Nothing else is checked, so nothing about the solution is revealed. Because the solution is unique, a full grid with no red marks is always the right one.

Mistakes - 3 lives. Every digit you enter is checked against the solution at once. A wrong digit is marked in red and costs a life; the third mistake ends the game and counts as a loss. Undo can take a wrong digit back, but not the life it cost.

Abandoned puzzles. Starting a new puzzle before finishing the current one counts as a loss in your statistics for that level.

Solving techniques

These are the methods the game's own solver uses to grade every puzzle, from the simplest up.

• The ends of the scale: the cell a sign points at can never hold the largest digit, and the other cell can never hold 1.
• A sign beside a known digit: if the larger side holds 3, the smaller side can only be 1 or 2.
• Singles: a digit with only one possible place in a row or column, or a cell with only one possible digit.
• Inequality chains: signs pass ranges along. In a chain a < b < c on a 5x5 grid, a can be at most 3 and c at least 3. Keep each cell's lowest and highest candidate in step with its neighbours.
• Pairs and triples: two cells of a row that can only hold the same two digits claim them; the rest of the row loses those digits.
• X-Wing: a digit confined to the same two columns in two rows can be removed from the rest of those columns.
• Hills and valleys: when two or more cells that share a row or column are all smaller than one cell - directly or through a chain of signs - they hold different digits, so the big cell must beat all of them. Two smaller cells in one row mean the big cell is at least 3, not just 2. The same works downwards for a "valley".
• Advanced: XY-Wing, Swordfish, quads and simple colouring - longer "if this, then that" reasoning, still without guessing.

Strategy tips

• Find the long chains. A run of signs pointing the same way pins its ends close to 1 and N.
• Mark 1 and N early. The smallest and largest digits are the most restricted: a cell with a sign pointing at it can't be N, a cell a sign points away from can't be 1.
• Write ranges, not just digits. Pencil the possible digits into chained cells, then tighten them from both ends.
• Use the rows too. Signs say what a cell can't be; the row says where each digit still has to go. Combining the two finds most singles.
• Look for hills. A cell with two smaller neighbours in the same row is at least 3.
• Don't guess. Every puzzle here can be solved by logic. If you feel stuck, look at a sign you haven't fully used yet.

Difficulty levels

The game generates every puzzle on the spot and grades it with a solver that works the way a person does, always using the easiest step available. A puzzle is given to you only if its hardest required step matches the level you chose.

• Easy: a 5x5 grid; comparing a sign with a known digit, the ends of the scale and singles are all you need.
• Normal: a 6x6 grid; at least one step needs an inequality chain or a pair.
• Hard: a 7x7 grid; at least one step needs a triple or an X-Wing.
• Crazy: a 7x7 grid; at least one step needs hills and valleys, an XY-Wing, a Swordfish, a quad or colouring.

Statistics and best times are kept separately for each level. If Futoshiki is new to you, start on Easy and move up once reading the signs comes naturally.