TWO RULE LAYERS
Layer 1 (Latin square): every digit from 1 to N appears exactly once in each row and column. On a 5×5 grid, each row and column contains 1, 2, 3, 4, and 5 in some order.
Layer 2 (inequalities): between some adjacent cells, a < or > clue tells you which cell must hold the larger digit. The clue is always satisfied — you cannot place a smaller digit on the > side.
In practice, the inequality clues do most of the early work. The Latin square constraint closes the puzzle once the inequalities have bounded most digit ranges.
TRACE THE CHAINS
A single < or > between two cells is one constraint. But clues often chain: A < B < C < D. This chain tells you:
- A is less than B, which is less than C, which is less than D.
- A must be at least 1 and at most N minus the chain length behind it. In a chain of 4, A can be at most N−3.
- D must be at least the chain length from the start. D ≥ 4.
On a 5×5 grid, a chain A < B < C < D forces A=1 and D=5 outright — only one value fits each extreme. The cells in the middle are constrained to {2,3,4} and resolve via the Latin square.
Look for the longest chain in each row and column first. Long chains are the highest-value clues in Inequality.
BOUND EVERY CELL'S RANGE
For every cell, collect all the inequality chains that pass through it (it might be in one chain horizontally and another vertically). Count:
- How many cells must be larger than this cell? Call it U (cells above in value).
- How many cells must be smaller? Call it L (cells below in value).
The cell's digit must be at least L+1 (to leave room below) and at most N−U (to leave room above). If L+1 = N−U, the digit is determined.
Example on a 5×5: if a cell has 2 cells that must be smaller and 2 cells that must be larger, its range is [3,3] — it must be 3.
CROSS WITH THE LATIN SQUARE
Once you've extracted everything the inequality clues give you, switch to Latin-square elimination: if a digit is already placed in a row, it cannot appear again in that row. Remove it from all other cells' candidate sets.
The interaction between the two layers is the engine of Inequality. An inequality bound narrows a cell to two candidates; Latin-square elimination then removes one; the determined digit satisfies an inequality that forces the cell next to it; which adds new Latin- square information. Follow the chain.
THE BEGINNER MISTAKE
Beginners read each < or > clue in isolation — "this cell is bigger than that cell" — without chaining. Isolated clues give only a direction, not a value range. Most of the time, a single two-cell clue doesn't force anything on its own.
The correct approach: always trace the full chain. Find the cell at the bottom of the chain (must be smallest) and the cell at the top (must be largest). Those extremes resolve first and propagate immediately. Experienced Futoshiki solvers count chain lengths before they look at individual clues.