Cyclic row-column shifts
Restore a numbered array using whole-row and whole-column cyclic shifts.
Shape & spatial thinking
- Players
- 1–4 cooperatively
- Time
- 15–25 minutes
- Materials
- Nine numbered cards, a 3×3 array and move log.
Prepare
The target rows, top to bottom, are 1,2,3 /4,5,6 /7,8,9. Label rows top-to-bottom R1–R3 and columns left-to-right C1–C3. There is no empty cell. The starter is 7,1,2 /3,5,6 /4,8,9. Copy it exactly and keep the target visible on a separate sheet.
Play
One move shifts all three entries of one row one cell left or right, wrapping the falling-off entry to the opposite end. Or shift one full column one cell up or down, also wrapping. Rows or columns never reverse order. Individual tiles may not swap places independently; lifting one card is only a physical aid to performing the entire declared cycle.
Worked example
In the starter, shift C1 up: the rows become 3,1,2 /4,5,6 /7,8,9. Then shift R1 left to reach the target. The second move acts on the already changed array. Writing only the moved strip without updating its three destination cells would lose information.
Finish and check
Finish when all nine positions match the target, then replay the written moves from a fresh starter to verify them. The supplied two moves prove solvability, not that every scramble needs two. For practice, a host may scramble the target with four declared legal shifts, keep their order privately, and check a solution by reversing both order and directions.
Adaptation and limits
Take turns as mover and recorder without scoring speed. Stop at a checked solution or fifteen moves. Use large number cards or write every state in a three-line log if manipulating cards is difficult. Do not import empty-square, sliding or pancake-flipping rules from neighbouring puzzles.
Purpose, observation and provenance
Within this activity: Restore a numbered array using whole-row and whole-column cyclic shifts.
Name a legal action, record the changed state and explain the check. One session does not measure lasting ability.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
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.
Rules to keep beside the materials
- The target rows, top to bottom, are 1,2,3 /4,5,6 /7,8,9. Label rows top-to-bottom R1–R3 and columns left-to-right C1–C3. There is no empty cell. The starter is 7,1,2 /3,5,6 /4,8,9. Copy it exactly and keep the target visible on a separate sheet.
- One move shifts all three entries of one row one cell left or right, wrapping the falling-off entry to the opposite end. Or shift one full column one cell up or down, also wrapping. Rows or columns never reverse order. Individual tiles may not swap places independently; lifting one card is only a physical aid to performing the entire declared cycle.
- In the starter, shift C1 up: the rows become 3,1,2 /4,5,6 /7,8,9. Then shift R1 left to reach the target. The second move acts on the already changed array. Writing only the moved strip without updating its three destination cells would lose information.
- Finish when all nine positions match the target, then replay the written moves from a fresh starter to verify them. The supplied two moves prove solvability, not that every scramble needs two. For practice, a host may scramble the target with four declared legal shifts, keep their order privately, and check a solution by reversing both order and directions.
- Take turns as mover and recorder without scoring speed. Stop at a checked solution or fifteen moves. Use large number cards or write every state in a three-line log if manipulating cards is difficult. Do not import empty-square, sliding or pancake-flipping rules from neighbouring puzzles.
Cyclic row-column shifts — play materials
| R/C | C1 | C2 | C3 |
|---|---|---|---|
| R1 | 7 | 1 | 2 |
| R2 | 3 | 5 | 6 |
| R3 | 4 | 8 | 9 |
| Target | C1 | C2 | C3 |
|---|---|---|---|
| R1 | 1 | 2 | 3 |
| R2 | 4 | 5 | 6 |
| R3 | 7 | 8 | 9 |
| Move1–15 | Row/column + direction | Resulting nine values |
|---|---|---|
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
Host check — keep private until finished
Starter 7,1,2 /3,5,6 /4,8,9 → C1 up →3,1,2 /4,5,6 /7,8,9 → R1 left → target.
These are model answers or legal examples, not records of an experiment or learner outcomes.
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