Understand fixed-step cycle coverage on a paper ring

A return can leave most labels untouched

Imagine twelve labels arranged around a paper ring, numbered clockwise from 0 through 11. Place a pencil beside label 0, then repeatedly move forward by four labels. Your landing sequence is 0, 4, 8, 0. You have returned to the beginning, but you have landed on only three labels. This is the distinction behind fixed-step cycle coverage: closing a route and visiting every permitted landing position are separate results. The example is a rule written on paper, not a description of a musical box's mechanism.

Here, a visit means a landing. If a drawn arrow from 0 to 4 passes the marks for 1, 2 and 3, those intermediate marks do not count as visited. That choice is part of the exercise. Without it, two people could follow the same drawing and report different coverage because one counted landings while the other counted points passed along the arc. Keep the landing rule beside the diagram before making either list.

You do not need to construct or run a physical carousel. A ring of handwritten labels and a separate sequence on paper are enough. The clockwise order supplies a convenient way to find the next label; it does not supply a measured speed, time interval, direction of a real motor or number of positions in a product. The twelve positions belong entirely to this invented example.

Write a rule that survives the wrap

Start at 0 and use increasing labels as the forward direction. Add the chosen step to the current label. If the result reaches twelve or more, subtract twelve to return to the available labels. For the steps used here, one subtraction is enough on each move. Thus moving four from 8 gives twelve, which becomes 0. Moving five from 10 gives fifteen, which becomes 3. These are bookkeeping operations on labels.

Record the starting position before making a move. After each addition and wrap, append the new landing. This produces one more written entry than the number of moves, because the initial entry was already there. The final return also repeats an earlier label. A list with four entries can therefore describe three moves and just three distinct landing positions. Count entries, moves and different labels separately.

A small recording sheet can use three columns: move number, current landing and whether that label is new. Enter move zero beside the starting label. When you return to the start, mark that entry as a repeat rather than crossing out the earlier entry. Keeping both makes the closing move visible while preserving an honest count of the labels encountered before the return.

Follow steps of four without filling the gaps

The first move takes 0 to 4. The second takes 4 to 8. The third takes 8 back to 0 after subtracting twelve. The complete first-return list is therefore 0 → 4 → 8 → 0. There are three transitions between four entries. The different labels are 0, 4 and 8. Labels 1, 2, 3, 5, 6, 7, 9, 10 and 11 have not been landing positions.

It might be tempting to continue the same rule in the hope that an overlooked label will appear later. The next move from 0 still goes to 4, followed by 8 and then 0. Nothing in the rule has changed. Every time the process reaches 0, it faces the same instruction it faced at the beginning. Repeating that three-move block adds entries to the record but adds no new landing label.

This last point follows from the specified rule, rather than from how long someone watches a moving object. The next position depends only on the current label and the fixed addition of four. The return puts the exercise in exactly the same state as before. If you allow the step to change, or allow a different direction on later moves, you have created another exercise and must write that change down.

To audit the result, make a second copy of the twelve labels in a straight row. Cross off a label the first time it appears as a landing. After the closing move, only 0, 4 and 8 should be crossed off. This independent checklist catches an easy error: calling the four written entries four different visits. The second 0 is necessary evidence of return, but it is not a fourth distinct label.

Compare a step of five on the same ring

Keep the same twelve labels and the same starting position, but replace the step of four with a step of five. The list begins 0 → 5 → 10. Adding five to 10 gives fifteen, so subtract twelve and land on 3. Continue the identical calculation from that new position. Do not restart at 0 merely because the arithmetic crossed the end of the label range.

The full first-return sequence is 0 → 5 → 10 → 3 → 8 → 1 → 6 → 11 → 4 → 9 → 2 → 7 → 0. Check each neighboring pair by adding five and subtracting twelve when needed. In particular, 8 leads to 1, 11 leads to 4 and 9 leads to 2. These wraps are part of the same rule as the straightforward move from 0 to 5.

There are twelve moves in that list and thirteen entries including the repeated final 0. Before the return, the different labels are 0, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2 and 7. Sorting that set gives 0 through 11 with no omission. This particular step therefore visits every label before returning. The conclusion comes from the complete enumeration, not from assuming that every closed route must have full coverage.

Lay the two first-return records side by side. Both close at 0. One covers three different labels and the other covers twelve. The return alone cannot distinguish their coverage, because it occurs in both records. A useful explanation needs the landing list or equivalent evidence of which labels appeared. A statement such as “it completed a cycle” leaves that central question unanswered.

Changing the start changes the question

Return to the step of four and start at 1 instead. The record becomes 1 → 5 → 9 → 1. Starting at 2 produces 2 → 6 → 10 → 2. Starting at 3 produces 3 → 7 → 11 → 3. Each run closes after three moves, but its landing labels differ from those of the run beginning at 0. Changing the initial label has not changed the step.

Across these four separately started runs, the combined landing sets contain all twelve labels: 0, 4, 8; then 1, 5, 9; then 2, 6, 10; then 3, 7, 11. That combined coverage is a result about four runs. It would be inaccurate to assign it to the single run beginning at 0. Keep the run boundary visible whenever a record includes a deliberate restart.

A new starting label does not necessarily introduce a new landing set. Starting at 4 with the same step gives 4 → 8 → 0 → 4, which contains the same three labels as the original run. It changes the order in which you first encounter them, not their membership. Before announcing extra coverage from another start, compare its actual labels with the ones already collected.

This example does not require a general rule for every ring size and every possible step. The claims here concern twelve labels, the two explicit steps and the listed starts. A reader can verify them directly. If you want to investigate another size later, first state its labels, wrap rule, start and visit definition, then build a fresh record rather than borrowing a total from this one.

Keep the surrounding objects out of the proof

The native PETSOLA Christmas carousel music box provides a visual association with a round decorative object. Its seven gallery images show a green and red carousel with Santa sleighs and reindeer, including detail views of the same item. A gallery measurement graphic conflicts with the written dimensions, so neither measurement is used here. Nothing in those views establishes twelve landing positions, a step of four or five, a measured rotation, or an observed tune.

The native PEPISKY horse-motif musical box is shown open in its single gallery image. The round box has a hinged round lid with a horse picture and a small central horse assembly. That complete object supplies another visual context, rather than a physical test. Its title does not establish a demonstrated lighting sequence, rotational behavior or audible result. The paper exercise remains independent of both products and needs neither purchase.

A neighboring explanation of choosing pairs for a display distinguishes six unordered selections from twelve left/right arrangements using four labels. Its question is which pairs count as different. The present question is which labels a repeated step reaches. Similar-looking lists do not make those questions interchangeable, and the number twelve in both examples does not provide a shared proof.

The display-code boundary example asks whether a string of marks can be decoded without ambiguous symbol boundaries. Here each ring label already has an explicit identity. We are following transitions between those labels, rather than reconstructing a message. A cleanly written sequence helps you audit the moves, but unique decoding and complete landing coverage remain different decisions.

If a separate purchase raises payment questions

If you separately consider a product on republeq.com and want to ask whether USDT could be used for that purchase, treat the answer as a checkout question for that particular order. This paper exercise does not establish that a listing, seller, account, location or checkout accepts any cryptocurrency. Read the options actually presented for the intended transaction, and ask the responsible support contact about anything that remains unclear before committing funds. The presence of a product link in an educational explanation does not answer those questions. If a cryptocurrency option is offered, establish which asset and network the checkout specifies, how the payable amount is determined, and what confirmation the order requires. Do not infer those details from another listing or an earlier transaction. Ask how cancellation and refunds would be handled for that order, including which currency would be returned and what conditions would apply. Keep the order information and the stated terms together so you can compare them with the action you are considering. If the asset, network, destination, amount or order association is uncertain, pause that transaction and obtain clarification rather than treating a guess as an instruction. Nothing in the ring diagram supplies payment directions or changes the terms of a purchase. You can complete every calculation here with paper and pencil, independently of any checkout choice. The two native listings serve only the limited visual context explained above. Their inclusion does not promise a discount, a particular conversion result, a fee level, a refund outcome or present eligibility to pay with USDT. Verify the actual order-specific terms if you choose to investigate purchasing after the exercise.

Report the result with its evidence

A concise record can say: with twelve labels, start 0 and a fixed forward step of four, the first return occurs after three moves and the landing set is {0, 4, 8}. With the same labels and start but a step of five, the first return occurs after twelve moves and the landing set contains all twelve labels. Each sentence states the inputs, return length and coverage, so another reader can check the claim.

Keep the original lists beside that record. They show why a return is evidence of a repeated state, while full coverage needs a separate accounting of the available labels. When a future drawing looks circular or repetitive, begin by asking what counts as a visit and which rule determines the next landing. In this paper example, those two choices make the difference visible before any total is announced.