Clock step cycles
Record a modular cycle and distinguish first return from visiting every position.
Adult planning suggestion: about 9–12 · Logic & rule checking
- Players
- 1–4 cooperatively
- Time
- 15–25 minutes
- Materials
- Circle labeled 0–11; two logs
Prepare
Label twelve positions 0 through 11, not 1 through 12. Put a marker at 0. A step adds a fixed integer and wraps by subtracting 12 as often as needed. Record the starting 0 as visit 0, before moving.
Play
Use step+4. Move and write each landing until the first repeated position. The sequence is 0,4,8,0. Stop at that return; it contains three distinct positions, not four. No marker or participant is removed.
Worked example
Restart at 0 on a fresh log with step+5. The landings are 0,5,10,3,8,1,6,11,4,9,2,7,0. Cross off each of the twelve labels on its first visit. All twelve are reached before returning.
Finish and check
A partner checks each adjacent difference modulo 12 and confirms that each run stops at its first return. Repeatedly writing extra rounds does not create extra distinct positions. For optional reverse motion use−4: from 0 the next label is 8, not a negative position.
Adaptation and limits
Finish after comparing the two distinct-visit counts 3 and 12. Learners may test another fixed step on a new sheet; state the starting label and circle size explicitly. This is a paper model of residues, not a claim about clock accuracy or time measurement.
Purpose, observation and provenance
Within this activity: Record a modular cycle and distinguish first return from visiting every position.
Ask learners to mark first visits separately from the repeated starting position and verify each wrapped difference.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
NRICH: Clock Arithmetic — Reference concerns modular arithmetic; our twelve-position comparison and logs are original.
Secondary practice: Cooperation and explaining a check.
For a class of 30, use ten groups of three. Prepare one player-material set per group and keep the host key with the adult. Rotate operator, recorder and checker after each worked example; a reader or scribe may help without choosing answers. Avoid public ranking or diagnostic labels.
Age bands are planning suggestions. Background references may address older learners and do not validate suitability for this age band. Goals describe observable actions, not measured long-term gains or institutional endorsement.
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
- Label twelve positions 0 through 11, not 1 through 12. Put a marker at 0. A step adds a fixed integer and wraps by subtracting 12 as often as needed. Record the starting 0 as visit 0, before moving.
- Use step+4. Move and write each landing until the first repeated position. The sequence is 0,4,8,0. Stop at that return; it contains three distinct positions, not four. No marker or participant is removed.
- Restart at 0 on a fresh log with step+5. The landings are 0,5,10,3,8,1,6,11,4,9,2,7,0. Cross off each of the twelve labels on its first visit. All twelve are reached before returning.
- A partner checks each adjacent difference modulo 12 and confirms that each run stops at its first return. Repeatedly writing extra rounds does not create extra distinct positions. For optional reverse motion use−4: from 0 the next label is 8, not a negative position.
- Finish after comparing the two distinct-visit counts 3 and 12. Learners may test another fixed step on a new sheet; state the starting label and circle size explicitly. This is a paper model of residues, not a claim about clock accuracy or time measurement.
Clock step cycles — play materials
| Visit0–12 | +4 | +5 |
|---|---|---|
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
Host check — keep private until finished
+4 has 3 distinct positions; +5 has 12.
These are model answers or legal examples, not records of an experiment or learner outcomes.
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