Rules
Renzoku, also called Consecutive, is a number grid. Fill each row and each column with the digits once: 1 through N, and N is the side. A 4×4 uses 1–4. The daily puzzle is a 5×5 and uses 1–5. Play runs from 4×4 to 9×9, each in easy, medium, and hard.
A dot between two cells means those digits differ by one. An empty gap means they do not. 1 and N are not consecutive, so they may sit side by side only when the gap is empty. Dots are on the sides, never on a diagonal. Some cells start filled. Those givens are bold and stay as they are. The grid is finished when every cell is filled, each digit appears once in every row and column, and every dot and every empty gap holds.
Sudoku is the same one-digit-per-line idea. Each row and each column — the only house on this board — holds every digit once. There are no boxes here.
- One of each digit. Every row and every column holds 1 through N once. N is the side. A 4×4 uses 1 to 4. A 9×9 uses 1 to 9.
- A dot, or an empty gap. A dot between two cells means those digits differ by one. An empty gap means they do not. 1 and N are not consecutive.
- Givens stay. A given digit is already filled, in bold. You can select it. You cannot change it.
- Finished. The grid is done when every cell is filled, each digit appears once in every row and column, and every dot and every empty gap holds.
In the top row, the dot sits between 3 and 4. The gap between 1 and 3 is empty, and the difference is 2. Every digit is teal, givens included. The dots stay ink.
- Dot: one apart
- Empty gap: not one apart
- Given
- Clash
Tap a cell. Arrow keys move the selection and stop at the edge. Type a digit or use the keypad. The pad shows how many of each digit are still to place. Erase, Backspace, Delete, or 0 clears the selected cell, digit and notes together. A given stays as it is. Notes (the Notes button, or N) are pencil marks. Shift with a digit toggles one note without turning Notes on. A note on a filled cell clears the digit first. When you place a digit, that digit leaves the notes in the rest of its row and column, in the same undo step.
The line under the keypad reads "Row R, column C · N cells to go", or "Given D · K of N on the board", or "Placed D · K of N on the board". Solved, the pad reads "Every row and column has each digit once, and every dot and gap holds." The dialog reads "Every row and column has each digit once, and every dot and empty gap holds."
A cell turns coral when it breaks a rule that is already visible: a repeated digit, a filled dot that is not one apart, or a filled gap that is one apart. Empty cells stay plain. A dot with one end empty is not a mistake yet. Press Hint for one digit the rules already force, with a sentence saying why. Press Check when you want the board checked. Undo and Redo step through the moves. Reset clears back to the givens. When the grid is complete and legal, the board turns teal, digits and frame. The dots stay ink.
- Undo
- Redo
- Notes
- Reset
- Hint
- Check
A dot or a gap
A dot beside a placed digit, an empty gap, one digit left in a cell, or a digit with one home in the row or the column.
A dot beside a placed digit
Start where a dot already touches a filled cell. The empty neighbour is one higher or one lower. The row and the column often remove one of those two. On a 4×4, a 1 leaves only a 2. A 4 leaves only a 3. 1 and 4 are not a dotted pair.
The blank 4×4 in the next section does not open here. Its first press is the empty gap, because the given sits on an empty gap and no dotted neighbour is filled yet. The press in this section is the next one. A 5×5 easy grid does open on a dot. The 5×5 easy grids do. There is no second grid for that sentence.
The 1 is already placed under a vertical dot. The cell above can be 2 or 4 before the dot speaks. 4 is not consecutive with 1. 2 is. Column 4 already holds the given 3, so 3 was already gone. Hint says "This dot touches a placed digit, so the neighbour is the next number up or down."
The next three presses on this board are the same kind of clamp, and they are not separate pictures. The cell left of the new 2, across an empty gap, becomes 4. The cell left of that 4, across a dot, becomes 3. The cell under that 4, across an empty gap, becomes 2. Then one digit is left in the first cell of the row.
No dot
An empty gap beside a placed digit forbids the number one higher and the number one lower. The other digits may still fit, until the row and the column have their say. No dot means not consecutive. It is not a blank you can ignore.
1 and N are not consecutive, so an empty gap beside a 1 still allows N. On this 4×4, an empty gap beside a 4 still allows 1. The check stays quiet until both cells are filled. The legal pair is in the mistakes section.
This is the first press on the blank board. The article meets it after the dot, because a dot is the earlier kind of step. The cell above the given 3 has no dot on that edge. Its list before the cut is 1, 2, or 4. 3 is already in the column. The empty gap removes 2 and 4. 1 remains, and 1 is not consecutive with 3. There is a dot under the given. That edge is a different rule. Hint says "There is no dot, so this neighbour is not next to the placed digit."
Only one digit left
Cross off every digit already in the row and the column. If one digit is left in the cell, write it. Hint says "Only one digit is left in this cell." It does not name the row. There is no separate sentence for the last empty cell of a line. When the cell is that last empty one, the button still says this.
The Sudoku guide already teaches this. The Sudoku sentence names the lines. This one does not. Do not look for a full line that fills itself under another name.
Row 1 already holds 3, 4, and 2, so the first cell is 1. The list was already a single digit before the press. The ring is that cell only. The rest of the row stays plain. Hint does not name the column.
The only home
A digit that can sit in only one cell of a row, or only one cell of a column, is written there. The cell itself may still allow other digits. Those other digits have homes elsewhere in the line. Hint says "This digit fits in only one cell of the row or column." It does not say whether it was the row or the column, and it does not give the line a number. The article may name the row. The ring is the cell.
The Sudoku guide teaches the same idea, and on that guide it is a medium step. Here it is one of the easy sentences. Not every easy grid uses it. The 4×4 in the earlier sections does not. This one opens on it.
The given 1 has an empty gap on each side, so those two cells lose 2 and keep 3 and 4. The first cell is not cut by that gap, because a dot sits between it and the next cell. 1 is already used in the row. Digit 2's only home in row 1 is the first cell. The column still allows 2 further down, so this is the row. The cell itself still allows 2, 3, and 4 until you write it. The ring is the cell, not the row.
That is the end of the easy sentences. An easy grid finishes with these four, and the path includes a dot or a gap. The next two sections are the medium look. A medium grid on the site can still be finished without the button ever saying them.
Two open cells
Two empty cells that share a side. A dot between them, or no dot, each drops a digit the other cell cannot match.
Two open cells and a dot
When two empty cells share a dot, each digit in one cell needs a neighbour one away in the other. If a digit has no such neighbour left, it leaves. 1 can only look for a 2. N can only look for N−1. The ends do not wrap. The partner does not have to be a single digit. One of its remaining digits is enough.
Hint says "These two open cells share a dot, so each digit needs a neighbour one away." This needs the medium or the hard setting. On an easy ceiling the button never says it.
The given 3 is in the bottom row, so the cell above the forcing dot can be 1, 2, or 4. The cell this step writes can be 2 or 4. A 4 there would need a 3 or a 5 below, and neither is left. A 2 still has a 1 below. The partner is not a single digit. The ring also includes the cell to the left, from an earlier look in the same pass. The forcing pair is the vertical dot. The pencil marks are the lists at that moment. They are not notes the solver reads.
Two open cells and no dot
With no dot between two empty cells, a digit is impossible once every digit the other cell can still be is one away from it. The usual picture is a cell whose two candidates are exactly the neighbours of the digit you want to cross off. Whichever way the partner goes, the empty gap would be consecutive. A partner that still has some digit farther away does not force the cut.
Hint says "With no dot, a digit is impossible once the other cell can only be next to it." The clean press sits on a 6×6, after sixteen cells are filled. The whole grid stays in the picture. A window would drop dots.
The dots above have already clamped the two empty cells. One can be 3 or 5. The other can be 2 or 4. There is no dot between them. 2 and 4 are exactly the neighbours of 3, so 3 leaves, and the cell is 5. 5 is consecutive with 4 and not with 2, so it stays. The two givens stay bold. Rows underneath are still empty. On this board the same press is available at medium. Easy does not say it.
That is the end of the two-cell look. Hard grids add a longer dotted run, a pair or a triple, and a one-digit try. A hard grid does not have to use all of them.
Runs and pairs
A straight run of dots has to be consecutive in one direction. Two cells, or three, can also claim the digits they share.
A dotted run
Three or more cells in a straight line, each pair joined by a dot, have to be consecutive in one direction or the other. On a 4×4 a run of four cells can only be 1-2-3-4 or 4-3-2-1. If one end cannot be 4, only one of those runs remains, and the other end is 1.
Hint says "This dotted run has to be consecutive in one direction or the other, and only these digits fit." This needs the hard setting. This board opens here.
The given 4 is in the top row, so the top of the run cannot be 4. The reverse run cannot start. The chain leaves 1, and the ring covers the whole column. Easy and medium do not say this on the blank board.
Two cells
Two cells in a row or a column that can only be the same two digits claim those digits. The digits leave the rest of the line, and a cell that was holding one of them may be filled. Hint says "These cells are the only home for these digits, so they leave the rest of the row or column." The same sentence covers three cells, in the next section. The picture shows two.
The Sudoku guide names the digits and the line. This sentence does not. A hidden pair, two digits that have only those two cells, uses the same sentence. The ring would be the two holders. No separate picture is drawn for that shape.
This is the first press on the board. A dotted run in column 2, inside the same pass, leaves two cells able to be 3 or 4. The button's sentence is the pair, not the run. The run already has its own picture. The dot beside the given 5 leaves the bottom cell as 4 or 6. The pair removes 4. 6 remains. The two cells of the pair stay empty, with the ring, so you can see the two-digit home. Easy and medium do not reach this on the blank board, because the run that builds those lists does not run under those ceilings.
Three cells
Three cells in a row or a column whose digits together are only three numbers claim those numbers. One of the three cells may hold just two of them. The numbers leave the rest of the line. The toast is the same sentence as the two-cell section. The picture shows three cells, plus the cell that gets the digit.
The Sudoku guide teaches the triple. There is no hidden triple here. Do not look for one. This needs a grid of at least 5×5, and the hard setting.
The first press on this board is an ordinary dot, the 2 beside the given 1. That dot is already taught on the 4×4. After the 2 is written, column 2 holds a 4/5, a 3/4/5, and a 3/4/5. Those three numbers leave the remaining cell, which was 1 or 3, and it is 1. The three claiming cells stay empty. If the puzzle is marked medium, the same cell is filled by the dot sentence instead, and the ring is only the dot. The figure's setting is hard.
What if
When the deductions above have not written a digit, pick an empty cell that has two digits left and try one. If that choice breaks a dot or a row, the other digit is the one that is written. Hint says "Trying this digit breaks a dot or a row, so the other digit is placed." The toast does not name the digit that failed. The digit on the board is the one that lives. The ring is only the written cell.
Both choices can also force the same cells. That sentence is in the solver, and it was not the button on the grids here. No picture. This needs the hard setting. Not every hard grid opens on a try. One opens on a run. One opens on a pair. This one opens here.
The empty gap above the given 4 has already removed 3, and 4 is used in the column, so the cell is 1 or 2. There is no dot between it and the cell above. Trying 2, that upper cell still holds 1 and 3. Both are one away from 2, and the empty gap forbids both, so the upper cell has nothing left. The button writes 1 and does not mention 2. The broken board is not drawn. Easy and medium return nothing on this blank grid.
That is the end of the hard sentences. A hard grid uses a run, a pair, a triple, or a try when the easier sentences are not enough. It does not have to use each of them.
Common mistakes
Check marks a cell only when a rule is already broken. A repeated digit is marked even when the row is unfinished. A dot is marked when both cells are filled and the digits are not one apart. An empty gap is marked when both cells are filled and the digits are one apart. A dot with one end empty is not marked. Notes are not marked, and they do not have to be cleared. 1 beside N on an empty gap is legal.
Hint, on the play page, refuses while Check would fail. The toast is "This board has a mistake, so there is no hint yet." The broken boards below show the coral cells and no amber ring. The one mistake that stays quiet is a digit that breaks nothing and is simply the wrong one. That figure shows the violet digit and the amber hint, on a different cell. Hint writes the forced digit and leaves the quiet one in place.
The 3 in the second cell is the digit that belongs there. The 3 in the first cell is not. There is no dot between them. The row still has empty cells. Check marks both 3s. Hint waits.
Same idea, down the first column. The two cells are in different rows. Neither 3 is the digit that belongs there. Check marks the column.
There is no dot on that edge, and the digits differ by 1. Both cells turn coral. The edge stays empty. Check says "Something breaks the rules. Look for the marked cells."
On a 4×4, 1 and 4 are the two ends. They are not consecutive, so an empty gap between them is legal. Both digits stay violet. Check stays quiet: "No mistakes so far. Keep going!" Hint asks for a different cell and does not clear these two.
The dot wants a difference of 1. These digits differ by 3. Both cells turn coral. The dot stays ink.
There is no dot between them, and 4 and 3 differ by 1. The given turns coral too, because it is one end of the broken gap. A 2 in that same cell breaks the gap as well. On this board a wrong digit in the first hint's cell always breaks a rule, so the Hint button refuses. It does not overwrite a quiet wrong digit there, because that cell has no quiet wrong digit.
The 2 repeats nothing, and no dot or gap beside it has both ends filled. It stays violet. No coral. The amber ring is the opening gap, on a different cell. If Hint runs, it places the 1 above the given 3 and leaves the 2 where it is.
What to play next
An easy grid finishes with a dot, an empty gap, one digit left in a cell, or a digit with one home in the row or the column. A medium grid adds two open cells that share a side, and it can also finish while the button speaks only the easy sentences. A hard grid adds a dotted run, two cells or three that claim a line, or one digit tried when the other breaks.
- EasyA dot, a gap, or one digitA dot beside a placed digit, an empty gap, one digit left in a cell, or a digit with one home in the row or the column. The path uses a dot or a gap.Play 4×4 Easy →Play 5×5 Easy →
- MediumTwo open cellsThose, and sometimes two open cells that share a side. A medium grid can also finish while the button speaks only the easy sentences.Play 4×4 Medium →
- HardA run, a pair, or a tryA dotted run, two cells or three that claim a line, or one digit tried when the other breaks. Not every hard grid uses each of those.Play 4×4 Hard →
4×4 through 9×9 each come in easy, medium, and hard. The daily puzzle is a 5×5 easy grid: today's Renzoku and 5×5 easy. A new player can start on the 4×4 easy grid, or on the daily.
Print a Renzoku pack. A single pack has eight puzzles. The starter pack has 18, one of every size and setting. Four 4×4, 5×5, 6×6, or 7×7 grids share a sheet. Two 8×8 or 9×9 grids share a sheet. Large print is one grid. The pages are black and white. Dots and bold givens print. Amber, red, teal, and notes do not.
Futoshiki uses signs between cells. Renzoku uses dots. Sudoku is the one-digit line, with boxes this grid does not have.
- GuideSudokuRows, columns and boxes, from singles to triples.
- GuideFutoshikiLess-than and greater-than signs between neighbours.
- GuideCalcudokuOne-way cages first, then the rest of the line.
- GuideHashiCount bridges, never cross, connect every island.
- GuideBinairoSpot pairs, fill the gaps, balance every line.
- GuideStar BattleOne or two stars in every row, column and region.
- GuideSlitherlinkDraw one loop. Start from the 0s and the 3s.
- GuideNonogramsOverlap long clues, cross the gaps, and read the crossing line.
- GuideHitoriShade repeats, never side by side, keep white connected.
- GuideNurikabeGrow each island to its number and keep one sea.
- GuideLight UpLight every cell. Start beside the 4s and 0s.
- GuideKakuroOne-way sums first, then short runs and longer lists.
- GuideKiller SudokuCage sums, then the rest of the line.
- GuideJigsaw SudokuFollow the thick walls. Each region has its own shape.
- GuideShikakuEach number is an area. Start with the ones that fit one way.
- GuideMasyuTurn on black pearls, go straight through white ones.