Futoshiki

Overview

Futoshiki is a Latin-square puzzle with inequalities between neighbouring cells. Every row and column must contain each allowed number once, and each greater-than or less-than sign must agree with the values beside it. A five-by-five board therefore uses one through five, with no Sudoku boxes and no arithmetic cage totals. Chains of inequalities make the puzzle more than a simple missing-number exercise. A cell smaller than three successive neighbours cannot hold a value near the top of the range. Boards often give no starting digits at all: the signs and the Latin-square rule alone supply every deduction. Smaller grids introduce the logic gently, while larger ones lengthen the chains and reward careful candidate tracking. The local Unequal engine generates puzzles and provides size and difficulty settings. You can record candidate numbers before committing them, then use the intersections of rows, columns and inequality chains to narrow the remaining possibilities.

How to Play

Click or tap a cell to select it, then enter a number with the keyboard or on-screen keypad. Use Backspace to erase and the arrows to move the selection. Secondary-select before typing when you want pencil marks. Read each inequality from the narrow end toward the wide end: the narrow end points to the smaller value. Fill all squares while satisfying every sign and avoiding duplicates in rows and columns. Pencil marks are worth using from the very first move, because a fresh board can hold several plausible candidates per cell until a chain resolves, and an early guess that contradicts a later sign is hard to trace without them. Undo reverses an entry. Restart keeps the current clues but clears your attempt.

Tips & Strategy

Look for long ascending or descending chains first. On a five-by-five board, a chain of five distinct increasing cells fixes all five values in order. Shorter chains still impose useful minimum and maximum values. Combine those bounds with numbers already placed in the same row and column, then inspect whether a missing number has only one possible square. Do not treat two cells without an inequality as ordered; they merely need to obey the Latin-square rules. When a candidate seems forced by one sign, follow the next sign in the chain before committing it. Also scan for cells squeezed from both sides, because a value caught between a larger and a smaller neighbour often collapses to one or two candidates immediately.

Reading Inequality Chains

A chain of signs behaves like a ruler laid across the board. On a one-to-five grid, a chain of three increasing cells means the first is at most three and the last is at least three, because the three values must be distinct and ascending. Lengthen the chain and the bounds tighten further: four increasing cells force the smallest to be one or two and the largest to be four or five.

Here is a concrete deduction. Suppose a cell X sits between two signs reading 4 > X and X < 3. The first sign says X is below four; the second says X is below three. The sharper bound wins, so X can only be one or two. If X's column already contains a two, the Latin-square rule finishes the job and X must be one. Notice the shape of the reasoning: one sign restricted the range, the second sign sharpened it, and the row-and-column rule supplied the final step. Train yourself to read every sign twice, once from each cell it touches.

Common Beginner Mistakes

• Treating two cells with no sign between them as ordered by habit. Without a symbol, either order is legal; only the row and column rules apply.
• Chasing an inequality chain so eagerly that a row ends up with a duplicated number. Every placement must satisfy both rule systems at once.
• Letting pencil marks go stale. After each confirmed entry, remove that value from the marks in its row, its column and the cells linked to it by signs.
• Forgetting that an inequality constrains the larger cell too. A sign narrows the candidates on both sides, and the high end of a chain is often where a board first cracks open.