How to solve Futoshiki: reading the signs

The rules

• On an n×n grid, every row and column contains the digits 1 to n once each.
• Every sign between two cells must hold: the open side faces the larger digit and the tip points at the smaller one.
• A few digits may be given at the start.

What one sign rules out

• The cell the tip points at is smaller, so it can't be the largest digit (n).
• The cell on the open side is larger, so it can't be 1.
• If one side is already known, the other is bounded: next to a 4 with the tip pointing at the empty cell, that cell is 1, 2 or 3.

Chains of signs

Signs often line up: a < b < c. A chain of length k forces its digits apart. In a chain of three increasing cells, the first can be at most n − 2 and the last at least 3. On a 4×4 grid, a chain of four cells along a row must be exactly 1, 2, 3, 4.

Chains can also bend around corners, through rows and columns. Follow every sign from a cell to see how far the chain goes: the longer it is, the narrower each cell becomes.

Combine with row and column logic

Once the signs have trimmed the candidates, Futoshiki plays like a Sudoku without boxes:

• If a digit can go in only one cell of a row or column, it goes there (a hidden single).
• If a cell has only one candidate left, fill it in (a naked single).
• Two cells of a row that can only be the same two digits remove those digits from the rest of the row (a naked pair).

Each new digit tightens the signs around it, which in turn creates new singles. Keep alternating between signs and lines.

A solving routine

1. Fill in the givens and cross them off their rows and columns.
2. Mark every cell next to a sign: no largest digit at the tip, no 1 on the open side.
3. Follow chains of signs and narrow each cell's range.
4. Look for singles and pairs in every row and column.
5. Re-check the signs around each new digit.

Practise

Puzzimo's Futoshiki comes in 4×4, 5×5, 6×6 and 7×7; the daily challenge is 5×5, and every puzzle has exactly one solution. The larger grids give only one to four digits and rely mostly on the signs. Notes, undo and hints are built in (a hint adds 30 seconds to a ranked time), and a wrong digit counts as a mistake; the third ends the round.

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