You already know how to solve a Futoshiki grid by applying simple exclusion (a digit can appear only once per row and column) and by eliminating candidates based on adjacent < and > signs? Here are some solving tips to help you crack tougher puzzles when those two basic techniques are no longer enough.
1. Naked pairs
If, within a single row or column, two cells contain only the same two candidates (for example, 2 and 4 in both cells, and nothing else), then those two digits must occupy those two cells—in one order or the other. We can therefore remove the 2 and the 4 from all the other cells in this row or column.
2. Hidden pairs
This is the mirror image of the naked pair. If, in a row or column, two digits can only be placed in the same two cells (even if those cells contain other candidates), then those two digits will necessarily occupy those two cells. We can therefore eliminate all other candidates from these two cells.
3. Naked triplets and hidden triplets
The same principle as pairs, but involving three cells and three digits.
- Naked triplet: three cells in the same row or column contain only three candidates among them (not necessarily all three in each cell, but no candidates other than these three digits). These three digits occupy these three cells, so they can be removed from elsewhere in the row or column.
- Hidden triplet: three digits can only be placed in these same three cells within the row or column, even if those cells contain other candidates. We then clean these three boxes of everything else.
The same logic applies to naked quadruplets (four cells, four digits); these are rarer but useful in larger grids (8×8, 9×9).
4. The X-Wing
A more subtle technique: if the same candidate digit appears in only two identical positions across two rows (for example, always in column 1 and column 6), these four cells form a rectangle. The digit will occupy either the “top-left to bottom-right” diagonal or the other diagonal—in either case, it occupies each of the relevant columns exactly once. We can therefore delete this figure everywhere else in these two columns.
Row 1: 4 is a candidate only in column 1 and column 4.
Row 4: here too, 4 is a candidate only in column 1 and column 4.
Whether the 4 goes in (R1,C1)+(R6,C6) or in (R1,C6)+(R6,C1), it will in any case occupy one cell in column 1 and one cell in column 4. We can therefore eliminate 4 from all other cells in these two columns.
5. Inequality chains
Beyond direct comparison between two adjacent cells, you need to learn how to follow a chain of signs that extends across several cells. The longer the chain, the more strongly it constrains the endpoints.
On a larger grid, a chain of length n does not necessarily dictate a single value, but it still eliminates the n-1 smallest candidates from the first cell and the n-1 largest candidates from the last. This is particularly effective when multiple chains share a cell: one can then combine the lower and upper bounds derived from each chain.
In what order should these techniques be applied?
These techniques can be combined with the already familiar Naked Pairs and Hidden Singles. In practice, it is most efficient to test them in order from simplest to most mentally demanding:
| Technique | Key indicator | Relative difficulty |
|---|---|---|
| Naked Pairs | 2 cells, 2 identical candidates | Moderate |
| Naked Triples | 3 cells, 3 candidates total | Moderate |
| Inequality chain | chain of < / > across 3 or more cells | Moderate |
| Hidden pairs | 2 digits confined to 2 cells | High |
| Hidden triplets / naked quadruplets | 3 or 4 confined digits | High |
| X-Wing | Rectangle across 2 rows / 2 columns | Very high |
Summary
When simple elimination and directly reading the < / > signs are no longer enough, first look for naked pairs and triplets, then their hidden counterparts, without forgetting to follow inequality chains across multiple cells. If the grid still resists, the X-Wing is often the key to unlocking the remaining candidates.
