How to play
Futoshiki is a logic puzzle from Japan; the word means “inequality”. On a board of size N, every row and every column must contain each digit from 1 to N exactly once, so the finished grid is a Latin square. A few cells start with a digit: these givens are bold on a grey background and cannot be changed. Between some neighbouring cells sits a sign. The signs < and > compare a cell with the one to its right, ∧ and ∨ compare a cell with the one below it, and the point of a sign always faces the smaller digit. Fill every empty cell so that no digit repeats in a row or column and every sign is true.
Choose a size from 4×4 to 9×9 and a level, then press Start. The clock stops when you pause, switch to another tab or finish. Your puzzle, notes, clock and undo history are saved in this browser, so you can carry on later.
Controls
- Mouse: click a cell, then type a digit or click it on the number pad.
- Touch: tap a cell, then tap a digit on the pad under the grid. The page does not scroll while your finger is on the grid.
- Keyboard: arrow keys move, 1 to N enter a digit, Backspace or Delete clears, N switches notes and Shift with a digit adds a note directly. U or Ctrl+Z undoes, Y or Ctrl+Y redoes, C checks, H gives a hint, P pauses and F toggles full screen.
- Notes: with Notes on, digits are pencilled in small as candidates. Placing a digit removes it from the notes in its row and column. Entering the same digit again clears the cell.
- Check and Hint: Check crosses out every wrong digit with a red line. Hint writes the correct digit into the selected cell if it is empty or wrong; otherwise it fills a cell that the basic rules or a pair can now decide, or else an open cell with the fewest options.
- Show conflicts: marks a repeated digit with a red “!” and rings a broken sign in red.
How it works
There are 576 different 4×4 Latin squares, 161,280 of size 5×5 and about 5.5 × 1027 of size 9×9, so a few signs and givens have to rule out an enormous number of fillings. Each puzzle here was made by a generator that started from a random complete square with every digit and sign shown, then removed clues one by one in random order. A removal was kept only if the puzzle could still be solved by logic alone, up to the rules allowed for its level. The page holds 1,440 such puzzles, 80 for every size and level.
The basic rule for a sign is simple. If cell a must be smaller than cell b, then:
Here max(b) is the largest digit still possible in b and min(a) the smallest still possible in a. The level is set by the hardest rule a puzzle needs:
- Easy: only the basic rules: a placed digit leaves its row and column, a digit with one possible cell in a row or column goes there, and the sign rule above. Easy puzzles also keep more givens and extra signs.
- Medium: needs at least one naked or hidden pair or triple. On 4×4, where pairs are almost never needed, Medium differs from Easy only by having fewer clues.
- Hard: needs an X-wing or a what-if test: assume one of the two candidates of a cell, follow the rules, and if that leads to a contradiction, the other candidate is right.
Each time you start, the page picks a puzzle you have not had recently and shows it in one of 16 versions: the 8 rotations and reflections of the square, each with or without every digit d swapped for N + 1 − d, which turns every sign around. Before the puzzles were stored, a separate brute-force solver confirmed that every one of them, in all 16 versions, has exactly one solution.
Tips and things to try
- Start at the signs: a cell that is smaller than a neighbour can never hold N, and a larger cell can never hold 1.
- Follow chains. In a row with a < b < c, a is at most N − 2 and c at least 3.
- Look for 1 and N in every row and column. They often have only one place left.
- Pencil in candidates with Notes and look for two cells in a line that share the same two digits.
- On Hard, pick a cell with two candidates and test one of them. If it breaks a rule, the other one is correct.
Frequently asked questions
Do I ever need to guess?
No. Every puzzle can be solved with the rules described above. The hardest step, on Hard only, is a short what-if test on a cell with two candidates.
What do the vertical signs mean?
∧ means the upper cell is smaller than the one below it, ∨ means the upper cell is larger. Just like < and >, the point faces the smaller digit.