Futoshiki Online

Futoshiki combines a Latin square with inequality clues

Futoshiki is a number-placement logic puzzle. A board of size N uses the numbers 1 through N, and each number must appear exactly once in every row and exactly once in every column. Between some neighboring cells, a less-than or greater-than sign adds another constraint. The value on the smaller side of the comparison must really be smaller. That combination creates a compact puzzle where row exclusions, column exclusions and inequality chains interact instead of acting as separate mini-games.

Start with the row and column rule before guessing values

On a 5×5 board every row and column eventually contains 1, 2, 3, 4 and 5 once each. If a row already has 1, 2, 4 and 5, its last empty cell must be 3 even when no inequality touches that square. The same reasoning works down columns. These Latin-square deductions are the backbone of the puzzle and provide exact candidates that can then be compared with the inequality clues around them.

An inequality restricts ranges as well as exact values

A sign does more than say which of two finished numbers is larger. If A is less than B on a 6×6 board, A can never be 6 and B can never be 1. If A is less than B and B is less than C, the chain gives all three cells tighter ranges before any one of them is solved. A must leave room above it, C must leave room below it, and B must fit between both. Longer chains make this range reasoning even stronger.

Horizontal and vertical signs describe the same relation

The browser board draws horizontal signs between left and right neighbors and rotates the same comparison mark between vertical neighbors. The logical meaning does not change with orientation: the open side faces the larger value and the pointed side faces the smaller value. Keeping one consistent interpretation matters when several comparisons cross a row and column, so the game highlights an inequality that is actually violated rather than relying on visual memory alone.

Candidate elimination is strongest when constraints overlap

Suppose a cell can be 2, 4 or 5 from its row, but it must be smaller than a neighbor that can only be 3 or 4. Values 4 and 5 disappear immediately, leaving 2. Conversely, a cell larger than a neighbor and missing only 1 or 3 from its column may be forced to 3. Efficient solving repeatedly intersects row candidates, column candidates and inequality ranges instead of treating any one source of information as sufficient by itself.

Four board sizes change the depth without changing the rules

V0.98 includes 4×4, 5×5, 6×6 and 7×7 play. Four by four is a fast way to learn how the signs and row uniqueness work. Five by five is the default because it creates meaningful chains without demanding a large screen. Six and seven by seven leave more candidates alive at once, so range deductions and cross-checking become more important. Every size uses exactly the same Latin-square and inequality rules.

Easy, Medium and Hard use separate frozen source pools

Every size has four Easy, four Medium and four Hard sources. Difficulty changes how much starting information is exposed and how densely the useful inequalities constrain the board; it never changes the rules or quietly gives a different kind of answer check. The result is forty-eight frozen source puzzles before symmetry transforms are considered. A Practice URL therefore selects from a real verified pool rather than inventing an unverified puzzle after the page loads.

Every source is counted to exactly one solution

The generator does not declare success just because its intended completed Latin square fits all clues. After clues are chosen, an exact backtracking solver counts solutions while enforcing row uniqueness, column uniqueness, immutable givens and every inequality. A source is shipped only when that count is exactly one. The independent release verifier repeats that work separately and also checks that the one solution it finds matches the frozen solution stored with the source.

Eight geometric variants expand replay while preserving the answer

Square grids can be rotated and reflected. The Futoshiki engine applies eight geometric symmetries to each frozen source, transforming givens, inequalities and the stored solution together. Forty-eight sources therefore produce three hundred eighty-four playable boards. The independent verifier checks every transformed board for uniqueness and confirms that its unique answer is the correspondingly transformed frozen solution, preventing a bad coordinate transform from silently changing puzzle meaning.

Practice becomes reproducible once its URL seed is fixed

Opening Practice without an s parameter creates a browser-local seed and writes it into the URL. That seed, along with the chosen size and difficulty, determines the source and symmetry variant. Reopening the same seeded URL selects the same puzzle; Fresh Practice creates another seed. This reproducibility is useful for sharing and QA without creating separate indexable pages for every board. The puzzle still runs entirely in the browser with no server-side generator.

Hint, Check and history have deliberately different roles

Hint fills one unresolved cell from the verified solution and marks the run Assisted. Check evaluates the current player state for row duplicates, column duplicates and broken inequalities without replacing the whole board with the answer. Undo and Redo restore exact player actions, including temporary conflicts that a person may legitimately create while experimenting. This distinction lets the tools support learning without pretending an assisted finish was the same as an unaided solve.

Autosave preserves the exact puzzle identity and givens

Practice autosaves under its size, difficulty and seed; Daily uses the calendar date. Returning to that identity restores the player grid, move count and assistance state. A saved player state cannot rewrite an immutable given, and the restore contract validates data shape rather than requiring every intermediate guess to already satisfy every rule. That means a temporarily conflicted board can be restored honestly instead of disappearing merely because the browser was closed at an inconvenient moment.

Clean and Assisted solves remain separate local facts

Using Hint, Check, Undo, Redo or Restart can reveal information or rewind a decision, so the run becomes Assisted. A completion still counts as a solved puzzle, but it cannot quietly replace a clean personal best. The statistics stored by the specialist are factual browser-local values such as solved puzzles, clean solves, best move counts by size and difficulty, and Daily history. There is no invented percentile, IQ estimate, global ranking or simulated online population.

The specialist avoids paid randomness and fabricated competition

All puzzle selection is deterministic and local. There are no chips, prizes, purchases, stakes or paid randomness, and no server opponent is implied. The frozen bank and solver are part of the project, so the game remains available without a paid API. This also makes quality assurance reproducible: a release verifier can regenerate or select the same puzzle, count the same solution set and compare the same transformed answer on another machine.

A practical solving loop is candidates, ranges, then chains

First mark the numbers missing from a promising row or column. Next remove candidates that are already present in the crossing line. Then apply nearby inequalities to remove values that are too high or too low to leave room for the required neighbor. When progress slows, follow a two- or three-sign chain and ask what minimum and maximum values each position can support. Repeating that loop usually creates a new single candidate without needing a speculative guess.

Daily Futoshiki stays separate from the frozen Today’s 6

The Daily mode derives one stable Futoshiki specification from the calendar date and records a specialist-only completion history and streak. It does not insert Futoshiki into the platform’s frozen Today’s 6 rotation, does not change the six-game Daily Score and does not manufacture a shared leaderboard. Players who want a new board at any time can use seeded Practice, while the specialist Daily remains a consistent once-per-date challenge.