Futoshiki Techniques: Reading the Inequalities

Updated June 2026

Futoshiki is what you get when you take a small Latin square and sprinkle greater-than and less-than signs between some of the cells. If you have ever done a Sudoku, half the puzzle is already familiar; the other half — turning those little arrows into hard numbers — is the part worth learning. This guide covers both, and the techniques that make a five-by-five board fall apart quickly.

The rules

• Latin square. On an N-by-N board, every row and every column must contain each number from 1 to N exactly once — no repeats in any line, just like Sudoku without the boxes.
• Inequalities hold. The signs between some neighbouring cells must be obeyed: the open mouth of the sign always points at the larger number. A cell on the tall side of the sign is greater than its neighbour on the other side.

That is the whole game. The skill is in noticing how tightly the two rules constrain each other.

The signs forbid the extremes

The fastest way into a board is to ask, for each cell touching a sign, which values it cannot hold. A cell that must be greater than a neighbour can never be 1 — there is nothing below 1 for the neighbour to be. Likewise, a cell that must be less than a neighbour can never be N, the largest value, because nothing exceeds it. On a five-by-five grid that alone often eliminates two or three candidates from the cells around every sign before you do any counting.

Follow the chains

Signs love to line up into chains, like a < b < c < d. A chain is a goldmine because it pins down ranges by position. The smallest cell in a rising chain of length k must be at least 1, the next at least 2, and so on — which means the largest cell in a length-k chain is at least k, and the smallest is at most N − k + 1. On a five-cell row, a chain of four rising cells is almost fully determined: the bottom of the chain can only be 1 or 2, the top can only be 4 or 5, and the middle values are squeezed hard. Always trace a sign to see how long its chain runs; the longer the chain, the more it tells you.

Combine the arrow with the line

The real power moves come from using an inequality together with the Latin-square rule, not on its own. Suppose a cell can only be a 4 or a 5 from row-and-column elimination, and a sign says it must be smaller than its neighbour. If the only value above it available in that column is also a 5, then your cell cannot be 5 (it must be strictly smaller), so it is a 4 — and the neighbour is 5. Each fact you nail down then propagates as an ordinary "this number is now used in this row and column" elimination, which is what keeps the solve rolling.

Look for the forced minimum and maximum

In every row and column, exactly one cell holds the 1 and exactly one holds the N. The signs frequently tell you where they cannot go. The 1 can never sit on the large side of a sign, so cross it off every "greater-than" cell; if that leaves only one cell in the line where a 1 is allowed, you have placed it. The same logic, mirrored, finds the N. Hunting specifically for where the 1 and the N must live is one of the most reliable ways to get the first few numbers on an otherwise blank grid.

A practical order of attack

1. For every sign, strike out the impossible extremes — no 1 on the big side, no N on the small side.
2. Trace each chain of signs and write down the tightened range for every cell in it.
3. Place any cell that now has only one candidate, then run the usual row/column eliminations from it.
4. Search each line for where the 1 and the N are forced.
5. Loop. Every new number tightens both its line and any sign it touches, so progress compounds.

Because each ujem Futoshiki board has a single unique solution, you never need to branch and guess — there is always a cell somewhere whose value is already decided by the signs and the lines together. Finding it fast is the whole sport in timed mode.