Half-turn region partition
Partition a grid into connected regions symmetric about their own fixed centres.
Logic & rule checking
- Players
- 1–3
- Time
- 15–25 minutes
- Materials
- Printed puzzle, pencil and separate host key
Prepare
Use the 4×4 grid with centres (1.5,2.5),(3.5,1.5),(3.5,3.5), where columns A–D have x=1–4 and rows 1–4 have y=1–4. A centre between cells is intentional, not a missing mark.
Follow the rule
Draw boundaries along cell edges so every cell belongs to exactly one connected region. Each region must contain exactly one marked centre. Connectivity uses shared edges.
Check each step
Rotating a region 180 degrees about its own centre must map every cell back into that same region. For cell (x,y) and centre (a,b), the required partner is (2 a−x,2 b−y). The rotation is a half-turn, not a reflection.
Worked example
For centre (1.5,2.5), A1 pairs with B4 and A2 with B3. One solution assigns the two left columns to this centre; use each right-hand 2×2 block for the other two centres.
Finish and adapt
Finish by testing partner membership, one centre per region, connectivity and full coverage without overlap. The sample is not claimed unique. A scribe can label regions P,Q,R instead of drawing heavy boundaries, provided the same edge partition is unambiguous.
Purpose, observation and provenance
Within this activity: Partition a grid into connected regions symmetric about their own fixed centres.
Check each constraint independently rather than compare only with the sample answer.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
Official mechanism reference / 官方机制规则 — Background rules only; all small boards, examples and explanatory text are prepared for this kit. No publisher artwork is reproduced.
Free alternative: Copy the supplied grids, labels and cards onto reused paper. No specialist product is needed. Adults may prepare pieces or act as a scribe.
Printable materials
Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.
Half-turn region partition · Player sheet
Before you begin
- Use the 4×4 grid with centres (1.5,2.5),(3.5,1.5),(3.5,3.5), where columns A–D have x=1–4 and rows 1–4 have y=1–4. A centre between cells is intentional, not a missing mark.
- Draw boundaries along cell edges so every cell belongs to exactly one connected region. Each region must contain exactly one marked centre. Connectivity uses shared edges.
- Rotating a region 180 degrees about its own centre must map every cell back into that same region. For cell (x,y) and centre (a,b), the required partner is (2 a−x,2 b−y). The rotation is a half-turn, not a reflection.
Dots may lie between cells. Partition along cell edges.
| Proposed cells / boundaries | Constraint checked |
|---|---|
| ________ | ________ |
| ________ | ________ |
| ________ | ________ |
Half-turn region partition · Host key — keep separate
P=A1:B4 (the two left columns); Q=C1:D2; R=C3:D4.
Read the rules as well as this example. Other valid solutions may exist for the puzzle boards. This key is not a claim of uniqueness or measured learning gains.
| Constraint / action | Evidence / correction |
|---|---|
| ________ | ________ |
| ________ | ________ |
| ________ | ________ |
| ________ | ________ |