Rotational Symmetry Puzzle Generator

Every cell in this grid belongs to exactly one region, and every region holds exactly one dot. A dot can sit in the middle of a cell, on the edge between two cells, or at the corner point where four cells meet — wherever it lands, spinning its own region 180 degrees around that exact point must land the region back on itself, cell for cell. No numbers appear anywhere on this puzzle: the only information is where each dot sits, and the only rule is that geometric symmetry. Every region must also hold together as one connected block, joined edge to edge rather than only at a corner.

Grid

Seed (optional)

The same seed rebuilds the same puzzle. Deterministic, not cryptographic.

A grid of dots, each waiting for its own symmetric region to be drawn around it, will appear here.

What this generator does

Carves the grid into regions one at a time from a shrinking pool of unclaimed cells: picks a random unclaimed cell, tries all nine dot positions touching it (its own centre, the four edges around it, the four corners), grows the largest region a flood-fill can support for each, and keeps the biggest. It repeats until every cell belongs to a region, then proves the resulting dots pin the grid down to that one partition and only that one, retrying with fresh randomness on the rare miss.

How to use this tool

Choose a grid size, then print or copy it.
Draw walls so every dot ends up alone in its own region.
Check that spinning each region 180 degrees about its own dot lands it back on itself exactly.
Keep every region joined edge to edge as a single connected block.
Show the answer to check the finished grid.

Understanding the controls

Grid: 5, 6, 7 or 8 squares a side. All four are measured well under a quarter of a second to build and prove unique, worst case.
Seed (optional): Type anything to rebuild the same puzzle later, which is how you reprint a page you have lost. The seed makes the puzzle repeatable; it carries no cryptographic strength and is not a secret.

Common use cases

A partition puzzle with no numbers anywhere, driven purely by a dot's exact position
A printable page that needs no key or legend, since a dot's own position is the entire clue
A geometry-flavoured change of pace from this site's numbered region puzzles
Reprinting the identical grid later from its seed

How this generator works

Two more obvious constructions were tried first and measured to fail. Scattering dots at random and growing each region outward independently left cells permanently boxed out by neighbouring regions in effectively every attempt — regions compete for the same cells, and once several stall the remainder can be unreachable by anyone. Splitting the whole grid recursively in half also fails, for a cleaner reason: splitting a region about a different point forces the untouched remainder to be the exact rotation of the kept half about the original point, which forces both halves to identical size — a region with an odd number of cells can never be split that way at all, and every offered grid here produces one sooner or later. What works, measured directly: carve regions sequentially rather than in parallel. Pick one unclaimed cell, try every dot position touching it, grow the largest region a flood-fill can support for that choice, remove it from the pool, and repeat. Nothing is competing for cells while one region grows, so nothing gets boxed out, and the pool only shrinks — the last few cells simply become small regions of their own. Measured over 60 seeded attempts at every offered size, this always covered the grid with connected pieces. Whether the finished dots are the *only* explanation is a separate question, checked by a full search that assigns cells to dots in pairs (assigning a cell forces its partner, by the same symmetry the puzzle relies on) and defers checking connectivity and coverage until a branch is complete. Measured over 60 seeded attempts per size, the dots were unique on the first try 100% of the time at five a side, falling to 88% at eight a side, so generation retries with fresh randomness on the rare miss — measured again over 150 attempts including every retry, the whole build never took more than 204 milliseconds, worst case, at eight a side.

Randomness and fairness

Which cell seeds each region, which of its nine candidate dot positions grows the largest piece, and the order regions are carved in, all come from your browser's cryptographic random source. Giving a seed replaces that source with a repeatable one, so the same puzzle can be printed again; the seed is for reproducibility, not cryptographic strength.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

Uniqueness on the first try falls as the grid grows, from 100% at five squares a side to 88% at 8 squares a side; nothing bigger was measured.
Difficulty is not graded — a grid can be settled from a handful of large dots or need several smaller ones puzzled out together.
Region sizes are whatever the carving happens to produce; a run of tiny one- or two-cell regions is possible, though the growth is biased toward the largest workable region at each step to avoid this being the norm.
There is no numeric clue anywhere on this puzzle; every deduction comes from a dot's exact position relative to the grid's own edges, which is a different kind of reasoning from this site's numbered puzzles and can take longer to get started with.

Privacy and your data

The region carving and the check that the dots are unique are both done entirely in your browser. Nothing you type as a seed, and no puzzle made from it, leaves the device.