Modular Lines Sudoku

Modular Lines Sudoku groups the digits by their remainder when divided by 3: {1, 4, 7}, {2, 5, 8} and {3, 6, 9}. Along a modular line the rule is that any three cells in a row must come from three different groups — one from each remainder class. It is the modular cousin of the entropic line, which groups by size instead of by remainder.

The rule: one remainder each, every three cells

Take any run of three consecutive cells on the line: it must hold one digit from {1,4,7}, one from {2,5,8} and one from {3,6,9}. Below, the colours mark the three remainder groups, and every three-in-a-row (1-2-3, 2-3-4, 3-4-5) has one of each.

A worked example, step by step

Take a line whose first two cells are a 1 and a 2, third cell empty.

Step 1 — identify the groups used. The 1 belongs to {1,4,7} and the 2 to {2,5,8}. Those are two of the three remainder groups, so the first three cells have already used them.

Step 2 — the third takes the last group. Only {3,6,9} is left, so the third cell must be a 3, 6 or 9.

The cycle repeats

As with entropic lines, the three remainder groups march in a fixed cycle along the whole line: once two neighbouring cells are grouped, every other cell's group is decided. That turns even a scattered pair of clues into a constraint on the entire line, which is why modular lines resolve so much of a grid at once.

Solving strategy

• Group before you place. Work out the {1,4,7}/{2,5,8}/{3,6,9} pattern along the line first.
• Two cells fix the cycle. Any two adjacent groups pin the rest of the line.
• Then use the grid. A cell narrowed to one group of three is easy for the rows, columns and boxes to finish.
• Feed the grid. Each digit fixed on the line behaves like a given for its row, column and box.

Common mistakes

• Checking only the first three. Every consecutive three must span all three groups, not just the opening trio.
• Grouping by size. Modular lines group by remainder (1-4-7 together), not by low/middle/high — that is the entropic line.
• Reading a group as one digit. {3,6,9} still needs the ordinary rules to pick the exact value.
• Forgetting ordinary Sudoku. Line cells still can't repeat within a row, column or box.

Can I play it here?

Not yet — Modular Lines Sudoku isn't one of the puzzle types SudokuStreak generates for now. But you can learn its rules above, and there's plenty here to sink your teeth into meanwhile:

• Browse all the Sudoku variants we do offer.
• Or print free puzzles from the printable PDF page.