Futoshiki puzzles

What it is

A small grid to fill with numbers like a miniature Sudoku, but with an extra twist: inequality signs between some cells telling you which neighbour must be larger, adding a layer of order to the logic. Futoshiki, meaning "not equal" in Japanese, is a logic puzzle in which you fill a square grid with numbers so none repeat in any row or column, while also obeying greater-than and less-than signs placed between certain cells. It blends the familiar Latin-square logic of Sudoku with the relational reasoning of inequalities, producing a compact but genuinely engaging deduction puzzle.

The rules are few and combine neatly. In an n-by-n grid, each row and column must contain the numbers one to n with no repeats, exactly as in a Latin square. Between some adjacent cells sit inequality signs, and the numbers in those cells must respect them, so a cell pointed at by the larger end must hold the bigger value. Some puzzles also give a few starting numbers. The interplay of the no-repeat rule and the inequalities is what makes each puzzle solvable by logic.

The inequalities are the distinctive solving tool. They constrain not just which numbers can go where but their relative order, so a chain of signs like one cell less than another less than a third forces a steep ordering that can pin down values quickly. The extreme values are especially revealing, since a cell that must be greater than several others is pushed toward the highest numbers, and one less than several toward the lowest, giving strong footholds.

It costs little, found in puzzle books and free apps, needs only a pencil, and suits anyone who enjoys Sudoku-style logic and fancies a fresh variation. The combination of familiar Latin-square deduction, the added dimension of inequality reasoning, and a compact format that still offers real challenge makes Futoshiki puzzles an elegant and rewarding mind-at-play pursuit.

How it works

Learn the two kinds of constraint and start small, because Futoshiki combines Sudoku's no-repeat rule with inequalities you must read carefully. Fill the grid so each row and column contains the numbers one to n without repeats, and make sure every greater-than or less-than sign between cells is satisfied, with the wider, open end of the sign pointing to the larger number. Begin with a small grid, perhaps four-by-four or five-by-five, in a book or app, and pencil in candidate numbers for cells as you would in Sudoku.

Exploit the inequalities for quick footholds. The signs are your most powerful tool: a cell at the small end of a sign cannot hold the highest value, and one at the large end cannot hold the lowest, which immediately eliminates candidates. Chains of signs are especially useful, since a run like one cell less than another less than a third forces a tight ordering. Combine this with the no-repeat rule, eliminating candidates that already appear in a cell's row or column, to narrow possibilities steadily.

Alternate between order logic and placement logic to finish. Solving flows by switching between the inequalities, which fix relative sizes, and the row and column constraints, which fix what is available, each narrowing the other. Use any given starting numbers as anchors, and look for cells where only one candidate survives both kinds of constraint. When stuck, recheck the inequality chains and the most filled rows or columns. A proper Futoshiki has a unique solution reachable by logic, so guessing is never required.

Read each inequality sign carefully, remembering the open end points to the larger value, since a single misread sign will send the whole deduction astray.

Benefits

Sudoku Logic Plus Inequalities Adds Relational Order Reasoning No Arithmetic Needed Compact but Genuinely Challenging Difficulty Scales Widely Screen-Free on Paper Cheap or Free to Play

What you need

Here's what to gather before you start. The essentials are marked.

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Futoshiki puzzles: from books or apps

A pencil: for entering and erasing numbers

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Pencil

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An eraser: since deduction involves corrections

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Eraser

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The two rules: no repeats, and inequality signs obeyed

Careful reading of the signs: open end means larger

A quiet moment: to concentrate on the deduction

Patience: to use the inequality chains well

FAQs

How is Futoshiki different from Sudoku?

It adds inequality signs to Sudoku-style logic. Like Sudoku, you fill the grid so each row and column contains the numbers one to n with no repeats, the structure of a Latin square. But Futoshiki also places greater-than and less-than signs between some cells, and the numbers must respect them, with the larger value at the open end of the sign. So where Sudoku relies purely on the no-repeat rule across rows, columns, and boxes, Futoshiki swaps the boxes for relational inequality clues, adding a dimension of ordering logic that gives it a distinct feel.

Do I need to do any maths?

No, only comparing sizes. Futoshiki involves no arithmetic, since you simply place numbers and check that they satisfy the no-repeat rule and the greater-than or less-than relationships. The only "calculation" is recognising which of two numbers is larger, which the signs demand. This makes it accessible to anyone who can count and compare, and the challenge is entirely logical: deducing values from the interplay of the inequalities and the row and column constraints. People who enjoy logic puzzles without sums often take to Futoshiki readily.

How do the inequality signs help me solve it?

They constrain relative size, which eliminates candidates fast. A cell at the small end of a sign cannot hold the highest value, and one at the large end cannot hold the lowest, so the signs immediately rule out possibilities. Chains of signs are especially powerful, since a run like one cell less than another less than a third forces a tight ordering that can pin down several values at once. Combining this order logic with the no-repeat rule for rows and columns is exactly how the puzzle resolves, making the inequalities the key solving tool.

Is guessing ever needed?

No, a proper Futoshiki has a single solution reachable by pure logic. You solve it by alternating between the inequalities, which fix relative sizes, and the row and column no-repeat rule, which fixes what numbers are available, each narrowing the other until cells resolve. Using any given starting numbers as anchors helps. If you get stuck, rechecking the inequality chains and the most filled rows or columns usually reveals the next forced placement. Guessing tends to introduce hard-to-unpick errors, so the logic alone should always carry you to the unique solution.