How Latin Squares Work

Summary: A Latin Square puzzle is a square grid: each symbol appears once in every row and once in every column. There are no boxes. This guide walks a 3×3, then the same two checks on a 5×5. Sudoku adds boxes on top of this rule. Word Squares is a different puzzle: a letter Latin square with one highlighted word.

What Is a Latin Square Puzzle?

A Latin Square puzzle is a square grid. Some cells already show a symbol. The rest are empty. Your job is to fill every empty cell so the finished grid is a Latin Square: each symbol appears once in its row and once in its column.

Sizes run from 3×3 to 9×9. A 3×3 uses the symbols 1, 2 and 3. A 5×5 uses 1 through 5. A 9×9 uses 1 through 9. The walkthrough starts on a 3×3, then uses the same two checks on a 5×5.

The left grid is the one to solve. The right grid is the finished key. Shaded cells were given. Count rows down from the top and columns left to right. The four steps below use the left grid only.

The Goal

Fill every empty cell so that each symbol appears once in every row and once in every column.

When every cell is filled and both checks hold, the page is done.

Latin Square Rules

Each row holds each symbol once. On a 3×3 that is 1, 2 and 3 in some order. A second 2 in the same row is wrong.
Each column holds each symbol once. Same check, read down.
There are no boxes. You do not hunt a 2×2, a 3×3 block, or any other region. A symbol may sit in the same 2×2 corner as another copy of itself, as long as they are not in the same row or the same column.
A given cell stays put. The printed symbols are part of the answer. Do not move them.
Every empty cell gets exactly one symbol. No blanks left. No two symbols in one cell.

That is the whole rule set. First look for a row or column that is only missing one symbol. If every line still has several gaps, find an empty that only one leftover still fits.

How to Play a 3×3 Latin Square

Work the left-hand 3×3. Use the two checks above.

Start with a row that is only missing one symbol. Row 1 is 2, empty, 3. The symbols for a 3×3 are 1, 2 and 3. 2 and 3 are already there, so the empty middle cell is 1. Write it.
Then read the column that just gained a symbol. Column 2 is now 1 on top, empty in the middle, and 3 at the bottom. 1 and 3 are taken, so the centre cell is 2.
Finish row 3. Row 3 is now empty, 3, empty. The leftover symbols are 1 and 2. Column 1 already has a 2 (row 1), so the left cell of row 3 cannot be 2. Write 1 on the left and 2 on the right.
The leftover middle row is forced. After step 2 the centre is already 2, so row 2 is empty, 2, empty. Column 1 still needs a 3; column 3 still needs a 1. Write 3 on the left and 1 on the right.

Check: every row is 1–2–3 in some order, and every column is too. That is a Latin Square. The two 3s in the bottom-left 2×2 are legal here — they do not share a row or a column.

How to Play a 5×5 Latin Square

The same two checks scale. A 5×5 uses 1 through 5. This 5×5 has no line that is only missing one symbol, so the first fill is an empty that only one leftover still fits.

The left grid is the puzzle. The right grid is the key. Shaded cells were given. Work the left grid.

This empty only fits one leftover. Row 5 is empty, 4, empty, empty, 2. The leftover symbols are 1, 3 and 5. Column 4 already shows 1 (row 1) and 5 (row 3), so of those leftovers the empty cell where row 5 meets column 4 cannot be 1 or 5. Write 3.
Then the empty in column 4 whose row already forbids one leftover. Column 4 still needs 2 and 4. Row 4 already shows given 2 and 1, so the empty where row 4 meets column 4 cannot be 2. Write 4.
Last symbol in that column. Column 4 is now only missing one symbol. 1, 3, 4 and 5 are taken, so the last empty (row 2) is 2.
Now the last two in row 5. Row 5 is now empty, 4, empty, 3, 2. Leftovers are 1 and 5. Column 3 already has 1 (row 4), so the empty cell where row 5 meets column 3 cannot be 1. Write 5. Row 5 is then only missing one symbol: write 1.
Last-symbol again on row 4. Row 4 is now 2, empty, 1, 4, empty. Leftovers are 3 and 5. Column 2 already has 3 (row 2), so the empty where row 4 meets column 2 cannot be 3. Write 5, then 3 in the last empty of row 4.

The rest of the page is the same hunt: an empty that only one leftover still fits, or a row or column that is only missing one symbol. Keep using those two checks until every row and column is 1–5. Match the left grid to the right-hand key when you want a check. A 5×5 with more empties takes longer; the two checks do not change.

Latin Squares with Letters, Pictures, or Colors

Digits are the easiest scan. The hunt is the same when the cells hold letters, Roman numerals, pictures, or colors instead of digits. Treat each distinct mark as one symbol. If a row already has a red cell and a blue cell on a three-color page, the last cell in that row is the leftover color.

Pictures take longer to scan than digits — match each image as one symbol, then run the same two checks.

Latin Square vs Sudoku

Sudoku keeps the Latin Square rule and then adds boxes. On a classic 9×9, each 3×3 box must also hold 1 through 9 once. A Latin Square page has no boxes. If the page itself is split into boxes, you are on a Sudoku page.

Word Squares is a different puzzle. It is a letter Latin square — each letter once per row and once per column, still no boxes — with one word highlighted across a full row or a full column. You fill the grid and read that word. This page is Latin Squares only: fill the grid.

Want a page of your own to try? Start with the free demo. To make a page in Puzzle Maker Pro, see How to Get Started with Puzzle Maker Pro – Latin Squares.

Outcome

You can fill a 3×3 by finishing any row or column that has one empty cell, start a 5×5 by finding an empty that only one leftover still fits, and tell a Latin Square from a boxed Sudoku page. More Latin Squares how-tos live in the Latin Squares tutorials. To print your own pages, see Puzzle Maker Pro – Latin Squares.

FAQ

What is a Latin Square puzzle?
An N×N grid that must hold each of N symbols once in every row and once in every column. Some cells are given. You fill the rest.

How is a Latin Square different from Sudoku?
Sudoku uses the same row-and-column rule and then adds boxes. A Latin Square has no boxes and no regions.

Can two of the same symbol sit next to each other in a Latin Square?
Yes, if they are not in the same row and not in the same column. Diagonal neighbours are fine. In Sudoku, the same pair is illegal only when both cells also share a box.

What sizes do Latin Squares come in?
3×3 through 9×9. A 3×3 is the shortest solve. A 5×5 is the default size. A 9×9 is the long page.

Do Latin Square puzzles have to use numbers?
No. The same rule works with letters, Roman numerals, pictures, or colors. Each distinct mark is one symbol, and that symbol still appears once per row and once per column.

Is Word Squares the same puzzle?
No. Word Squares is a letter Latin square with one highlighted word on a row or a column. Latin Squares is the grid only.