Renzoku Puzzle

Renzoku is a number puzzle, also called Consecutive. You fill a square so each row and each column contains every digit from 1 through N once. A dot between two cells means those digits are consecutive. You can play grids from 4×4 up to 9×9, each in easy, medium, and hard. Every puzzle here has exactly one solution.

Rules

The board is an n×n grid. Each cell is its own square. A narrow gap separates neighbours. A thick frame sits around the outside. N is the side of the grid, so a 5×5 uses the digits 1 through 5.

A dot is drawn in the gap, centred on the shared edge. It is part of the puzzle, not a cell you fill. A dot means the two digits differ by exactly one.

Where you see no dot, the two digits are not consecutive. That is a rule, not a blank you can ignore: no dot means NOT consecutive. Both directions count. A pair with a dot that is not one apart is a mistake. A pair with no dot that is one apart is a mistake as well. The check looks at a pair only once both cells are filled.

The ends do not wrap. 1 and N are not consecutive. On a 9×9, 1 and 9 are not a dotted pair. On a 4×4, 1 and 4 are not a dotted pair.

Each digit appears once in every row and once in every column. A repeated digit in a row or a column is a mistake even when the line is still unfinished.

A given is a digit already printed in the cell. You cannot change it. Your own digits are the ones you type. Pencil notes are small marks inside an empty cell. They are not the answer, and they do not have to be removed when the grid is finished.

The puzzle is solved when every cell holds its digit and no rule is broken. Notes are ignored.

Controls

Tap a cell to select it. The arrow keys move the selection, and they stop at the edge of the board. With a cell selected, tap a digit on the pad or press that digit on the keyboard. The pad shows how many of each digit are still to place.

Press N, or the Notes button, to toggle pencil marks. In that mode a digit writes a small note instead of filling the cell. Shift plus a digit writes a note even when Notes is off. A note on a filled cell clears the digit first. Pressing the same note again removes it.

Backspace, Delete, 0, or Erase clears the selected cell, digit and notes together. A given stays as it is. An empty cell with no notes stays put.

A cell turns coral when it breaks a rule that is already visible: a repeated digit, a filled dot that is not consecutive, or a filled gap that is consecutive. Press Hint for one digit the rules already force, with a sentence saying why. Press Check when you want the board checked against the rules. Undo and Redo step through your moves, and Reset clears the board back to the givens. When the grid is complete and legal, the board turns teal.

Techniques

Easy grids follow from one placed digit:

• Dot: a dot next to a placed digit leaves only the number one higher or one lower.
• Gap: an empty gap next to a placed digit forbids the two neighbours of that digit.
• Single: a cell that has only one digit left is filled with it.
• Hidden: a digit that fits in only one cell of a row or column is written there.

Medium grids look at two open cells that share a side. A shared dot means each digit needs a neighbour one away. An empty gap forbids a digit once the other cell can only be next to it. Hard grids use a longer dotted run, a pair or a triple that claims a row or column, or try one digit and keep what both choices force.

Beginner tips

Start with a dotted pair that already has one digit. The empty cell is then one higher or one lower, and the row and column will often drop one of those two.

When a gap has no dot, cross off the consecutive digits before you look for a place to write. Remember that no dot means NOT consecutive, and that 1 and N are not consecutive, so the two ends can sit side by side when the gap is empty.

If you are new to Renzoku, start with a 4×4 easy grid, or the daily 5×5 easy grid, and move up once the dots and the empty gaps feel natural. Play runs from Easy → Hard on every size from 4×4 to 9×9.