gamesblog

Latin squares

A small grid where every row and column holds 1 to n exactly once. Find the number in the marked cell, from 3×3 up to 7×7, or fill the whole square.

  • Logic
  • Untimed
  • 6 levels

How it plays

Every row and every column of the square must hold each number from 1 to n exactly once. Some cells are given; work out the number in the cell marked ?.

Look at the marked cell’s row and column together: the numbers they already hold are ruled out. When more than one is left, settle a neighbouring cell first — often a row with one gap, or a number that has only one place left in a column.

Levels grow the square from 3×3 and take away more of the given numbers, down to as few as still leave a single answer. Every puzzle has exactly one completion, checked by trying them all. You can also choose to fill the whole square.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises constraint reasoning — ruling out until one value is left — on Latin squares. It claims nothing wider.

Learn

How to get better at latin squares: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Read the ? cell’s row, then its column, and list every number the two hold between them. Those are ruled out.
  2. If one number from 1 to n is left, that is the answer. Say the list out loud — “row has 2 and 4, column has 1” — so nothing is double-counted or missed.
  3. If two or more are left, settle another cell first. Look for a row or column with one gap, or a number that has only one possible place in the ? cell’s row or column.
  4. Feed what you settled back into the ? cell and rule out again. On the hardest squares this can take two or three rounds.

Techniques and tricks

  • Row plus column

    The ? cell can only take a number missing from both its row and its column. Combine the two lists before comparing with 1 to n.

    e.g. A 4×4 row reads 2 ? 4 _ and the ? cell’s column has a 1: 2, 4 and 1 are out, so ? is 3

  • The last gap

    A row or column with one empty cell is finished by the one number it lacks. Fill such lines first; each one adds a number to some other line.

    e.g. A 5×5 column holding 5, 1, 4 and 2 has a 3 in its gap

  • Only one place

    Pick a number and a line, and ask where in that line the number can go. If every cell but one is blocked — taken, or the number already sits in that cell’s other line — it goes there.

    e.g. The 3 is missing from column 2, and rows 1, 3, 4 and 5 already hold a 3, so column 2’s 3 is in row 2

  • Settle the blocker

    When the ? cell has two candidates, look at the cells that share its row or column and could take one of them. Settling one of those often removes a candidate from the ? cell.

  • The line total

    Each row and column adds to the same total: 6 on a 3×3, 10 on a 4×4, 15 on a 5×5, 21 on a 6×6, 28 on a 7×7. A quick check on a nearly full line, or on a filled square.

    e.g. A 6×6 row with 1, 2, 4, 5 and 6 adds to 18, so the gap is 21 − 18 = 3

As the levels climb

  • The ladder starts on a 3×3 with about seven tenths of the cells given, and the ? cell settled by its row and column alone. Then the square grows to 4×4.
  • Next, fewer cells are given — about half — and the ? cell is one its row and column do not settle on their own: you must work out another cell first. Then the square grows to 5×5.
  • Then the givens go down to as few as still leave one answer, and the square grows to 6×6 — or to 7×7 if you raise “Largest square”. The ladder ends there.
  • If Time pressure is on, a marked cell gets 15 seconds plus 3 for each empty cell; filling the whole square gets 30 seconds plus 8 for each.

Traps to avoid

  • Checking only the row, or only the column. A number the other line already holds then looks possible, and those numbers are among the options.
  • Picking a number both lines still allow. On the harder levels the first wrong options are exactly those: row and column both leave them open, and only another cell rules them out.
  • Stopping at two candidates and guessing. On the harder levels the ? cell usually has more than one candidate at first; the step that settles it is somewhere else on the board.
  • Misreading which row or column the ? sits in on a big square. Put a finger on the ? and trace across and down before you list anything.

How to practise

  • Switch “Answer” to “Fill the whole square” once single cells come easily. Filling every cell makes the last-gap and only-one-place moves into habits.
  • Raise “Largest square” to 7 when the 6×6 squares with the fewest givens fall without a guess.
  • Raise “Options shown” towards the size of the square, so every number is on screen and you cannot rule options out by eye.
  • When you miss a harder one, read the full square in the working and find the cell that should have been settled first.

Levels

It is untimed unless you turn the clock on. There are 6 levels with the default settings, and the counter stops at the real top. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More logic

All games →