Futoshiki

How to play Futoshiki

Futoshiki is a number-placement puzzle played on a square grid — 4×4, 5×5 or 6×6 in this version — where the aim is to fill every cell with a digit so that each row and each column contains every digit from 1 up to the size of the grid exactly once, just like a Latin square. What sets Futoshiki apart from other number puzzles is that some pairs of neighbouring cells also carry a "<" or ">" inequality sign between them, and the two numbers on either side of that sign must satisfy it. The name comes from the Japanese for "inequality", and the puzzle is a favourite of logic-puzzle fans because it blends the row-and-column discipline of a Latin square with an extra layer of comparison reasoning. A handful of starting numbers, called givens, are already printed on the board; everything else you have to work out using nothing but the rules and the inequality signs.

The goal

Fill in every empty cell so that each row and each column contains the digits 1 through N (where N is the size of the board — 4, 5 or 6) exactly once, with no repeats, AND so that every visible "<" or ">" sign between two neighbouring cells is satisfied by the numbers you place there. The puzzle is solved the moment the whole grid is filled in and every rule holds at once. There is nothing to add up and no guessing required — every puzzle on this site is generated with a solver that proves it has exactly one solution before you ever see it, so patient logic alone is always enough to finish it.

The rules

• Every row must contain each of the digits 1 to N exactly once — no digit may repeat anywhere in the same row.
• Every column must also contain each of the digits 1 to N exactly once — no repeats down any column either.
• Some pairs of cells that sit directly next to each other — either side by side in a row, or one above the other in a column — have a "<" or ">" sign printed between them. The two numbers that end up in those two cells must make that sign true.
• Read the sign the same way you would read it in maths: the open (wide) side of the sign always faces the larger number, and the pointed (narrow) tip always points at the smaller number. For example, if the sign between two side-by-side cells is "2 < 5", the left number must be smaller than the right number.
• A few cells are already filled in when the puzzle starts — these are the "givens", shown in a different colour. They are fixed clues and cannot be changed; your job is to work out the value of every other cell around them.

A worked example

Picture two cells side by side in the same row with a "<" sign printed between them, so it reads left-cell < right-cell. If the left cell can only be 3 or 4, and the right cell can only be 3 or 4 as well, the sign tells you immediately that the left cell must be the smaller of the two possible numbers and the right cell the larger — so the left cell is 3 and the right cell is 4. Inequality signs like this are often the fastest way into a puzzle: whenever a sign sits between two cells whose possible numbers barely overlap, work out which cell has to take the smaller value and which has to take the larger one, and you will often pin down both cells at once.

Controls

• Tap (or click) an empty cell to select it. The cell you picked, along with every other cell in its row and column, is highlighted so you can see at a glance what it interacts with.
• Tap a number button below the board — labelled 1 up to the size of the grid — to drop that number into the selected cell. On a keyboard you can simply press the matching number key.
• Made a mistake? Tap Erase, or press Backspace/Delete, to clear the selected cell. Given clues are locked and cannot be erased or changed.
• The inequality signs between cells are shown automatically wherever they apply — there is nothing to toggle. Just remember that every visible sign must hold once both of its cells are filled in.

Difficulty levels

• Easy plays on a 4×4 board — the smallest grid, with the fewest digits to juggle and the shortest chains of reasoning. It is the best place to learn how the inequality signs interact with row and column logic.
• Medium plays on a 5×5 board. There is more room for numbers to hide, so you will need to track several rows, columns and inequality chains at the same time.
• Hard plays on a 6×6 board — the biggest grid here, with the most digits, the most inequality signs to track and the deepest reasoning chains. Every puzzle, at every size, is still guaranteed to have exactly one solution.

Solving tips

• Start with the inequality signs, not the empty cells. A chain of two or three connected signs (for example a < b < c) often rules out the smallest and largest possible numbers immediately, because a cannot be the biggest number in the grid and c cannot be the smallest.
• Cross off candidates using both rules at once. Before guessing, list which numbers a cell cannot be because of its row, which it cannot be because of its column, and which it cannot be because of any inequality signs touching it — often only one number survives all three checks.
• Look for cells at the "end" of a long inequality chain, since they are the most restricted. A cell that is smaller than three of its neighbours can never hold the largest few digits on the board, and a cell that is larger than three neighbours can never hold the smallest few.
• Work outward from the givens. Fixed clue numbers immediately rule out that value everywhere else in their row and column, which is often enough to narrow a nearby cell down to a single legal candidate.

Daily puzzle

Tap Daily to load that day's special puzzle for the selected difficulty. Everyone who plays sees the exact same grid on the same UTC calendar day, so you can compare notes and times with other players, or challenge yourself to beat your own daily best. The Daily puzzle changes automatically at midnight UTC and is generated the same fair way as every other puzzle — always with exactly one solution.

Scoring and the leaderboard

Each solved puzzle earns a score of max(1, min(99999, 10000 − seconds × 5 − mistakes × 200)). You start from 10000 points; every second on the clock costs 5 points, and every wrong number you place costs 200 points, with the final score always kept between 1 and 99,999. So the faster and cleaner your solve, the higher your score. Scores are tracked separately for each difficulty — sign in to save your best result to the leaderboard and see how you compare with other players around the world.

Frequently asked questions