How to Solve Futoshiki: A Step-by-Step Guide for Beginners

If you've found a Futoshiki puzzle and you're staring at a grid full of little greater-than and less-than signs wondering where to begin, this is for you. Futoshiki is far simpler than it looks: fill the grid so no number repeats in any row or column, and make sure every arrow between cells points from the bigger number to the smaller one. There's no math, no guessing, and the arrows actually make it easier than a blank sudoku, because they hand you relationships to work from. This step-by-step guide walks you through your first solve with a full worked example.

For the formal rules and the symbol reference, see the Futoshiki rules page or the visual how-to-play tutorial. This is the friendly, hands-on version.

What you're actually looking at

A Futoshiki puzzle is an N×N grid — usually starting at 4×4 — with two kinds of clues:

• A few given digits already placed.
• Inequality signs (<, >) sitting between some pairs of adjacent cells.

Your goal is to fill every cell with a digit from 1 to N so that:

• Every row and column contains each digit exactly once (the Latin square rule — same as sudoku, minus the boxes).
• Every inequality is satisfied — the open mouth of the sign always faces the larger number, so 3 > 1 and 2 < 5.

That's the whole game. No arithmetic, just order and placement.

Step 1: Read the arrows for "can't be" rules

Before placing anything, use the arrows to rule values out. In a 4×4 grid (digits 1–4):

• A cell with an arrow saying it's greater than a neighbor can't be 1 — there has to be something smaller than it.
• A cell that's less than a neighbor can't be 4 — there has to be something larger.

These "can't be" facts are free, and they stack up fast. Mark them mentally or as pencil notes before you do anything else.

Step 2: Find the forced ends of chains

Look for runs of arrows pointing the same way, like a < b < c. The values must increase along the run. In a 4×4 grid, a three-cell increasing chain can only be (1,2,3), (1,2,4), (1,3,4), or (2,3,4) — and if it's a four-cell chain, it can only be 1,2,3,4 in order. The longer the chain, the fewer ways to fill it. Solve the longest chains first; they often hand you several digits at once.

Step 3: Anchor off the given digits

Use any given number together with its arrows. If a cell shows 2 and an arrow says its neighbor is smaller, that neighbor must be 1. If the neighbor is larger, it's 3 or 4 — and the Latin square rule (checking the row and column) usually trims that to one option. A single given digit at the end of a chain can cascade values right down the line.

Step 4: A full worked example (4×4)

Picture the top row of a 4×4 Futoshiki with this clue between the first two cells: A > B, and suppose cell C in that row is given as 4.

• Since C is 4, neither A nor B can be 4 (no repeats in the row).
• A > B means A isn't the smallest, so A isn't 1, and B isn't the largest available, so B isn't 3 (the top remaining value). With 4 taken, A and B come from {1, 2, 3}, and A > B.
• The only pairs from {1,2,3} with A > B are (2,1), (3,1), (3,2). Now check the column constraints: suppose B's column already contains a 1. Then B can't be 1, so B = 2 and A = 3.
• That fills A = 3, B = 2, C = 4, leaving the fourth cell of the row as 1 by elimination.

A whole row solved from one arrow, one given digit, and the no-repeat rule — no guessing anywhere. String enough of these together and the grid fills itself in.

Step 5: When you get stuck

Every beginner hits a wall. The fixes, in order:

• Re-read every arrow near the stuck area. You've probably not fully used one — especially the "can't be 1 / can't be N" rules.
• Look for the longest chain you haven't finished. Its ends are the most constrained cells.
• Write candidate lists. Jot the possible digits in each empty cell, then prune by row, column, and arrow.
• Look for naked singles. A cell where the row, column, and arrows leave only one digit is solved, even if it didn't look obvious.

And never guess. Our puzzles always have exactly one logical solution, so a coin-flip feeling means there's an inequality you haven't squeezed yet.

Step 6: What comes next

Once forced extremes and short chains feel natural, learn the technique that makes Futoshiki click: mastering inequality chains. Then put it all together with the full Futoshiki strategy guide. For bigger boards, see how the puzzle scales in Futoshiki grid sizes.

Where to play your first one

Open an easy 4×4 Futoshiki — the small grid and generous arrows make it the gentlest place to learn. Prefer paper? Grab a printable Futoshiki. The best way to learn how to do a Futoshiki puzzle is to solve one: read the arrows, place a forced cell, and go.

Frequently asked questions

How do you do a Futoshiki puzzle?

Fill the grid so every row and column contains each digit from 1 to N once, and so every inequality sign is satisfied (the open side faces the larger number). Start by using the arrows to rule out values — a cell greater than a neighbor can't be 1, a cell less than a neighbor can't be N — then work the chains and given digits.

Where do you start in a Futoshiki puzzle?

Start with the arrows. Mark which cells can't be the smallest or largest value, then solve the longest inequality chains, which collapse to very few arrangements. Anchor off any given digits and let their values cascade along the arrows.

Is Futoshiki solvable by logic alone?

Yes. A well-made Futoshiki always has a single solution reachable by pure deduction — you never need to guess. If you're stuck, there's an inequality or a chain you haven't fully used. Re-read the arrows around the area that won't budge.

Does Futoshiki involve math?

No. Despite the greater-than and less-than signs, there's no arithmetic — you never add or calculate anything. The signs only tell you which of two cells is larger. It's pure ordering and placement logic, like sudoku with relationship clues.

What do the symbols in Futoshiki mean?

The < and > signs between cells show which neighbor is larger: the open, wider side always faces the bigger number, and the point faces the smaller one. So 2 < 5 and 5 > 2 mean the same thing. The full symbol reference is on the rules page.