A Futoshiki grid looks simple: a square of empty boxes and a scattering of "less than" and "greater than" signs. But those little arrows change everything. Instead of just filling in numbers so each row and column has no repeats, you also have to satisfy every inequality on the board. Get one arrow wrong and the whole puzzle unravels a few moves later.
This guide is for anyone who has stared at a Futoshiki grid, placed a few numbers, then hit a wall with no idea what to try next. Maybe you already solve Sudoku and want a new logic challenge, or you tried Futoshiki once and got stuck on the symbols. Either way, we will start from zero and build up real technique.
By the end, you will understand the rules, know how to mark candidates without cluttering your grid, and have a set of concrete strategies — from intersection elimination to reading inequality chains — that turn guesswork into deduction. We will finish with a full 5x5 puzzle solved step by step, plus tips for handling bigger, tougher grids.
What Is Futoshiki and Why the Symbols Matter
Futoshiki was invented in 2001 by Japanese games designer Tamaki Seto, and the name gives away the whole game: futoshiki (不等式) means "inequality" in Japanese. English-language editions sometimes call it "More or Less" or "Unequal," which is an even clearer hint at what you're doing. You'll find it in the Saturday Guardian, the Daily Telegraph, and most days (except Saturday) in The Times and the Dundee Courier.
The grid works like a Latin square, similar in spirit to Sudoku: each row and column must contain every digit exactly once. What sets Futoshiki apart are the < and > signs placed between certain adjacent cells. These aren't decorations — they're binding constraints. Every arrow must point from the smaller number to the larger one, and satisfying them alongside the row and column rules is the entire puzzle.
The Basic Rules: Latin Squares Plus Inequalities
Futoshiki runs on two rule sets stacked together, and you need both satisfied at once.
First, it's a Latin square. On a 5x5 grid, every row and every column must contain the digits 1 through 5 exactly once. No repeats, no skips. If you've solved a Sudoku row before, this part will feel familiar.
Second, the inequality signs between certain adjacent cells (< or >) must hold true in the finished grid. A "<" between two cells means the left or top value is smaller than the neighbor it points away from; the arrow always points toward the smaller number.
Here's the key habit to build early: treat these as one combined constraint, not two separate puzzles. A move that satisfies the Latin square rule but breaks an inequality is wrong, and vice versa. Every number you place has to pass both tests simultaneously.
Start With Pencil Marks
Before you commit to a single number in any cell, write down every value that could still work there. On a 5x5 grid, that means jotting small 1s through 5s in the corners of each empty box. This habit—pencil marks, or candidate notes—is widely considered the single most important tool for cutting down mistakes in Futoshiki.
Here's why it matters: Futoshiki mistakes rarely come from bad logic. They come from forgetting that a number is still possible somewhere else in the row. Pencil marks fix that by keeping every option visible.
A simple workflow:
- Scan each row and column, cross out numbers already placed.
- Check inequality signs touching the cell, remove values that would break them.
- Update marks immediately whenever you place a new number.
This small bit of upfront bookkeeping makes every later technique—including intersection elimination—much easier to spot.
Use Intersection Elimination to Place Easy Numbers
Once you've marked candidates, look for numbers that are almost fully placed across the grid. If a digit already appears in every row but one, you know it has to land somewhere in that leftover row. The question is where.
This is where columns do the work. Check which columns already contain that digit — those columns are ruled out for the missing row. Whatever column is left standing is the answer.
Here's the process in practice:
- Scan the grid for a number that's missing from just one row.
- Note which columns already have that number placed.
- Eliminate those columns from the empty row's candidates.
- If only one column remains open, place the number there immediately.
This technique is satisfying because it doesn't require guessing or deep logic chains — just careful cross-checking. It's often the fastest way to crack open a stalled grid, and it pairs well with your pencil marks, since you can visually confirm the overlap before committing to a placement.
Spot Inequality Chains
A single inequality sign only compares two cells, but strings of them together create a chain — and chains are where the real deductions live.
Look for patterns like a < b < c < d across a row or column. On a 5x5 grid, that chain tells you immediately that "a" can't be anything but 1 or 2, and "d" is almost certainly a 4 or 5, since three other values have to fit beneath it in ascending order. The longer the chain, the tighter the squeeze.
Work both directions:
- Count how many cells sit below a symbol in the chain — that's the minimum value that cell must clear.
- Count how many sit above it — that's the maximum value it can be.
A four-link chain in a 5-cell row or column often solves two or three cells outright before you write a single pencil mark elsewhere. Scan every row and column for these runs first; they're the fastest shortcut to hard numbers early in the puzzle.
Narrow Candidates With AB Elimination
Once pencil marks are in place, look for two cells in the same row or column that share the exact same pair of candidates and nothing else. If cell one can only be 2 or 4, and cell two — in that same row — can also only be 2 or 4, you've found an AB pair.
Here's why that matters: those two values must occupy those two cells, in some order. It doesn't matter yet which cell gets which number, but no other cell in that row can be a 2 or a 4. You can erase those candidates everywhere else in the line.
This technique works because a Futoshiki row is a closed set — every digit from 1 to the grid size appears exactly once. Locking two values into two cells shrinks the possibilities for everyone else.
Practical steps:
- Scan each row and column for cells with identical two-candidate pencil marks.
- Confirm no third cell shares that same pair.
- Strike those two digits from every other cell in the line.
Small eliminations like this often cascade into full solves.
Worked Example: Solving a 5x5 Futoshiki Step by Step
Picture a 5x5 grid, empty except for a few inequality signs between adjacent cells. Numbers 1 through 5 fill each row and column exactly once.
Start by writing small pencil marks 1-5 in every cell. This is tedious but pays off fast.
Say the top-left cell has a "greater than" arrow pointing into it from the cell to its right: left cell > right cell. That single sign eliminates 1 as a candidate for the left cell (nothing is smaller than 1) and eliminates 5 from the right cell (nothing is larger than 5). Cross those off in your pencil marks.
Now look for intersection elimination. If the digit 3 already appears in four of the five rows, scan the columns for where 3 is still possible in the fifth row. Often only one column has 3 left as a candidate, so you can place it immediately.
Next, hunt for chains. Suppose you have a > b > c across three cells in a row. That chain alone forces a into the higher half of the range and c into the lower half — a can't be 1 or 2, and c can't be 4 or 5. Update pencil marks accordingly.
With most cells narrowed to two or three candidates, apply AB elimination. If two cells in the same row both show only "2 or 4" as options, no other cell in that row can hold a 2 or a 4. Strip those digits from every other cell's marks.
Repeat this cycle — pencil marks, intersections, chains, AB pairs — and the grid tightens row by row until every cell holds a single confirmed number.
Tips for Tackling Larger and Harder Boards
Bigger grids don't require new logic, just more patience. A 7x7 or 9x9 Futoshiki has more candidates per cell, so your pencil marks matter even more — skipping them is where most solvers get stuck.
A few habits help as boards grow:
- Work the edges first. Corners and border cells have fewer neighbors, so inequality chains resolve faster there.
- Track one number at a time. Instead of solving cell by cell, scan the whole grid for every possible "3," then every possible "7." This mirrors the discipline Queens players use when checking one color region across the whole board before moving on.
- Expect sparse clues. Harder puzzles often give fewer inequality signs, forcing you to lean on Latin-square logic (row/column exclusion) rather than chains alone.
- Re-scan after every placement. A single new number can unlock several dormant deductions elsewhere.
Larger boards reward systematic habits, not cleverness — the same logic just takes more repetitions.