THE SETUP
Futoshiki (Japanese: "not equal") is played on an N×N grid — typically 5×5 or 7×7. Like Sudoku, every row and column must contain each digit from 1 to N exactly once. The difference: instead of pre-filled cells, the puzzle places < and > signs between adjacent cells in the same row or column.
Some cells may also be pre-filled with given digits — these work exactly like Sudoku givens. But the defining characteristic of Futoshiki is the inequality signs. A sign pointing from cell A to cell B means the value of A is less than the value of B.
GridJoy calls this puzzle "Inequality" — same mechanics, more descriptive English name. If you've played Futoshiki elsewhere, you'll recognise every rule immediately.
WHY INEQUALITY SIGNS ARE DIFFERENT FROM GIVEN DIGITS
A given digit in Sudoku tells you exactly one thing: the value of one cell. An inequality sign tells you something about two cells simultaneously — their relative order. This is both less information (you don't know either value directly) and potentially more information (the sign constrains both cells at once).
The key: inequality signs constrain the range of values each cell can hold. If a cell has the sign "< neighbour" and the grid is 5×5, that cell cannot be 5 (it must be less than something). If a cell has "> neighbour" for two different neighbours, the cell must hold a value larger than both — which quickly narrows the candidates to the top of the range.
THE EXTREMES LOCK FIRST
The fastest entry into any Futoshiki grid: count the inequality arrows per cell to find the minimum and maximum bounds.
- A cell with arrows pointing away from it on all sides (it's greater than all neighbours) must hold a large value. If it's greater than 2 neighbours in a 5×5 grid, its minimum value is 3.
- A cell with arrows pointing toward it on all sides (it's less than all neighbours) must hold a small value. Greater than 0 neighbours means it can be as small as 1.
- In the extreme case: a cell that is greater than every one of its neighbours must hold the maximum value (N in an N×N grid). A cell less than every neighbour must hold 1.
These extremal cells are immediate forced placements — find them first before doing anything else. They give you concrete digits the same way givens do.
CHAINING INEQUALITIES
The real power of Futoshiki comes from following chains of inequality signs across a row or column. If cell A < cell B < cell C, then A must hold a value at least 2 less than C, and C must hold a value at least 2 more than A. In a 5×5 grid, a chain of three cells A < B < C forces A ∈ {1, 2, 3}, B ∈ {2, 3, 4}, C ∈ {3, 4, 5} — three cells each narrowed to three candidates before you've touched anything else.
The longer the chain, the stronger the constraint. A full chain A < B < C < D < E across a 5×5 row forces the entire row: A = 1, B = 2, C = 3, D = 4, E = 5. One chain of four signs solves an entire row instantly.
Always scan for the longest chains first. They provide the most information per sign and often bootstrap the entire solve.
HOW IT COMPARES TO SUDOKU
Both puzzles use the same Latin-square base: each row and column holds each digit exactly once. The solving vocabulary overlaps — naked singles, hidden singles, candidate elimination all work in Futoshiki.
The difference is the information source. Sudoku gives you direct values (given cells). Futoshiki gives you relational constraints (inequality signs). The relational constraints require a different first step: deriving bounds from chains rather than eliminating candidates based on given placements.
Experienced Sudoku players typically find Futoshiki accessible within a session or two. The chain-tracing skill is new; the elimination reasoning once bounds are established is identical to Sudoku.
HOW IT COMPARES TO SKYSCRAPER
Both Futoshiki and Skyscraper add one type of relational clue to a Latin-square grid. Skyscraper places visibility-count clues outside the grid; Futoshiki places comparison signs inside it.
Futoshiki's signs are more local — each sign relates exactly two adjacent cells. Skyscraper's clues span an entire row or column (the view from the edge). This makes Futoshiki more granular: a sign's effect is immediate and specific, while a Skyscraper clue requires reasoning about all N cells in a sequence at once.
Most players find Futoshiki easier to enter than Skyscraper because the local constraint is more concrete: "this cell is bigger than that one" is simpler to reason about than "exactly 3 buildings visible from this side."
THE SOLVE ORDER
- Find the longest inequality chains across rows and columns — apply the chain bounds immediately (minimum value = chain position from the small end + 1; maximum value = N − chain position from the large end).
- Find extremal cells (greater than all neighbours → value = N; less than all neighbours → value = 1). Place them.
- Apply given digits (if any) and propagate via row/column elimination — same as Sudoku naked singles.
- Use updated candidates to re-check inequality signs: if a cell's minimum candidate exceeds its neighbour's maximum, that inequality can be used to eliminate further candidates.
- Repeat until solved. Most beginner and medium Futoshiki grids resolve in 2–3 passes without guessing.