Consecutive Sudoku Rules: How Bars Change Every Move

Consecutive sudoku is normal sudoku plus a line of white bars: two cells joined by a bar must hold consecutive digits, and two cells without a bar must not. That second half of the rule does more work than most players expect. We generated six puzzles with our engine — two easy, two medium, two hard — solved each one, and treated every adjacency in the solutions as a potential bar. Across 864 adjacencies we found 224 real bars. Here is what they taught us.

Consecutive Sudoku Rules in One Paragraph

The nine rows, nine columns, and nine 3x3 boxes all behave exactly like normal sudoku. On top of that, every pair of side-by-side cells (up, down, left, right) is either joined by a bar or left bare:

• A bar means the two digits must differ by exactly one: 1-2, 5-6, 8-9. Same digits are banned, and gaps of two or more are banned.
• No bar means the opposite — the two digits may not be consecutive. In standard notation every bar is shown, so an empty gap is a constraint, not an absence of information.

Two edge facts drop straight out of the definition. The digit 1 has exactly one consecutive partner, 2; the digit 9 has exactly one, 8. Any bar that touches a 1 or a 9 locks the other end the instant you see it. And because 4, 5, and 6 each have two partners, a bar touching one of those digits only narrows the other end to two options — the middle digits are the flexible ones.

Real talk: beginners treat the bars as decoration and solve the grid as plain sudoku. The bars are not decoration. They are the strongest single-source constraint on the board, and they fire before you write your first candidate.

What 224 Bars Taught Us: The Measurement

Every solved 9x9 grid offers 144 side-by-side pairs. Our six solutions carried between 31 and 45 bars each — a mean of 37.3, or 25.9% of all adjacencies. Pure chance puts the number at 22.2%, so a solved grid naturally produces roughly one barred pair in four. Puzzle makers do not invent bars from thin air; they select from the bar set the solution already contains.

| Difficulty | Clues | Bars in solution | |:-----------|------:|-----------------:| | Easy 1 | 40 | 45 | | Easy 2 | 40 | 40 | | Medium 1 | 34 | 35 | | Medium 2 | 34 | 33 | | Hard 1 | 30 | 31 | | Hard 2 | 30 | 40 |

Then we applied the rule to the puzzle state, one pair at a time, before any solving logic:

• 224 barred pairs measured; 179 of them (79.9%) left at least one end with exactly one possible digit.
• 156 pairs (69.6%) actually cut candidate lists on first application, removing 1.45 candidates per pair on average.
• The bar rule alone pinned 334 cells to a single digit across six puzzles — 55.7 per puzzle.
• 150 bars had a given on one side, and 133 of those (88.7%) pinned the empty partner immediately. The partner averaged 1.11 candidates left — usually one, occasionally two.

The no-bar rule ran quieter and still mattered: across 640 unbarred pairs it removed 160 candidates, an average of 0.25 per pair. Small per pair, but the pairs outnumber bars three to one, and every removal is a digit you never have to test.

Push both rules to a fixpoint — keep applying them until nothing changes — and 385 cells across the six puzzles (64.2 per puzzle) end up with exactly one candidate. On easy grids that is the whole solve.

Two Worked Examples

Here is the strongest bar-and-given case from our hard puzzle (30 clues):

Row 7: 0 0 8 0 1 5 0 0 0
Row 8: 2 0 0 0 0 0 0 0 4

The cell at row 7, column 1 is empty with candidates {3, 4, 6, 7}. A bar connects it downward to row 8, column 1, where a 2 sits. The bar says the pair must differ by one, and only one candidate satisfies that: 3 sits next to 2, while 4, 6, and 7 do not. Four candidates collapse to one on the strength of a single given.

The no-bar rule gets its own moment in the same grid:

Row 6: 0 9 0 0 2 6 0 1 0

The cell at row 6, column 7 carries candidates {3, 5, 7, 8}. The 6 to its left has no bar between them, so the pair may not be consecutive. That bans 5 and 7, leaving {3, 8}. Two candidates gone without a single pencil mark — and the 3 turns out to be correct.

How to Solve Consecutive Sudoku: Order of Attack

The numbers point at a clear sequence:

• Hunt barred pairs with a given first. 88.7% of them pin the empty partner on contact. When the given is a 1 or a 9, the partner is forced outright; when it is a middle digit, the partner drops to two options.
• Then sweep the no-bar rule next to every given. A fixed digit bans its two neighbors from all surrounding cells that lack a bar. This is the quiet worker — 160 eliminations in our six grids, most of them cheap.
• Feed every placement back into the bars. Each new digit turns nearby bars into instant pins and nearby gaps into double bans. The fixpoint is not a theory; our propagation pass pinned 64.2 cells per puzzle, and on the easy grids that single pass solved almost everything.
• Watch the 1 and 9 edges. They are the only digits with one partner, which makes every bar they touch a forced move.

Real talk: if you solve consecutive sudoku like plain sudoku, you are throwing away the highest-value constraints on the board. The bar rule pins something on four out of five contacts. Start your scan at bars with filled cells — they are the fastest wins in the variant, and each one pays into the next.

Try It Yourself

Consecutive sudoku rewards the same candidate thinking you already practice here. Watch single candidates fall in the hints tool, warm up on a full board in the step-by-step sudoku guide, and generate fresh puzzles at your level in the maker. When you want more room to stretch the habit, 16x16 mega sudoku runs the identical logic on a bigger canvas.