Rotating paper insert compatibility: keep gap order

State what the paper insert must match

Rotating paper insert compatibility needs more than the number of markers around a circle. In this original fictional exercise, an insert has four markers and the target has four marked sites. Even matching their collection of gap lengths does not establish a fit. The clockwise order of those gaps can prevent every allowed rotation from registering the two patterns.

Imagine two ideal paper diagrams with the same fixed centre and radius. Each has twelve equally spaced sites numbered zero through eleven clockwise, with zero at the top. The insert may rotate by any whole number of site steps. It may not move its centre, change size, stretch, flip over or relocate individual markers. These are stipulated rules, not instructions for handling an actual component.

A match requires the insert's four marked sites to occupy exactly the target's four marked sites after one allowed rotation. The markers are indistinguishable, so no colour or individual name must match. Unmarked sites remain unmarked. There is no physical thickness, clearance, tolerance or fastener in this model, and a successful calculation supplies no finding about a manufactured insert.

Keep the native canvas accounts separate

The native U.S. Art Supply round canvas account names a 30-inch two-pack. Its sole image shows two round white stretched canvases with a matching diameter and pack label. Those are attributed offer details, not independent measurements or authentication. The canvases provide art-workspace context; they are not the insert and target used in the fictional registration exercise.

The native chaize assorted canvas account explicitly describes 24 canvases, six each in square, round, hexagon and heart shapes. All five gallery images were reviewed, including the blank-shape panel, integrated stretcher back, painting demonstration and finished-artwork room scenes. The demonstration's paint and tools are illustrative context, not verified supplied inventory. No material, priming, certification, performance or actual insert-fit claim is established here.

Calculate a successful turn without changing the pattern

Call the insert pattern S. Its marked sites are zero, one, four and six. Call the first target T, with marked sites two, three, six and eight. A turn of two steps clockwise sends zero to two, one to three, four to six and six to eight. The resulting four sites are exactly T's four sites.

For these inputs, two steps is therefore an allowed matching offset. Each marker moves by the same amount. Moving the marker at four by two steps while leaving the others fixed would change the pattern rather than rotate it. Keep the common offset beside the result so the calculation shows which permitted operation produced the match.

Site numbering wraps after eleven. A marker at ten moved four steps clockwise would visit eleven, zero, one and then two. Thus its final site would be two. This separate counting illustration defines the numbering convention; it does not add another marker to S or change the successful offset of two in the original comparison.

Inspect every allowed offset once

Twelve whole-step offsets, zero through eleven, cover the allowed turns. An offset of twelve returns every site to its starting position and repeats offset zero. For each offset, move all four marked sites, wrap their numbers into the zero-through-eleven range and compare the resulting set with the target. Order in the written set is irrelevant to membership.

For S, offsets zero, one and two produce the sets {0,1,4,6}, {1,2,5,7} and {2,3,6,8}. Offsets three, four and five produce {3,4,7,9}, {4,5,8,10} and {5,6,9,11}. The third set in this list is the successful comparison with T, corresponding to offset two rather than a third-step turn.

Offsets six, seven and eight produce {0,6,7,10}, {1,7,8,11} and {0,2,8,9}. Offsets nine, ten and eleven produce {1,3,9,10}, {2,4,10,11} and {0,3,5,11}. These six results include the wraparound cases. Together with the first six, they complete the twelve-offset check without inserting or removing a marker.

Only offset two yields T. The complete check therefore establishes both a match and its uniqueness among the permitted whole-step turns for these particular inputs. A first successful comparison would establish existence, but it would not by itself show that no second permitted orientation also works.

Test a target with the same marker count

Now introduce a different target, U, marked at zero, one, three and six. Retain the original S and all the same rules. U still has four markers, the same centre and the same stipulated radius. Those agreements are necessary parts of the model, but they do not settle whether an allowed turn maps S onto U.

Compare U with each of the twelve resulting sets above. None is {0,1,3,6}. At offset zero, S has a marker at four where U instead requires three. Choosing a different offset moves all four markers together; it cannot move only that one marker left while keeping the other three fixed.

A shorter exhaustive check is also possible here. S has a marker at zero, so a successful rotation must send it to one of U's marked sites: zero, one, three or six. Those possibilities restrict the candidate offsets to zero, one, three and six. Their resulting sets were already listed, and none equals U. Every other offset fails immediately because S's zero marker would land on an unmarked target site.

This reasoning concerns the explicitly allowed whole-site turns. It does not authorize cutting, stretching or modifying a real insert to force a match. Under the declared agreement, U is incompatible with S. Changing the agreement would create a separate question that should keep its own inputs and result.

Keep the clockwise gaps in order

Starting at S's marker zero and walking clockwise, the next marker is one step away. From one to four is three steps; from four to six is two; and from six back to zero is six. S's ordered gap sequence is therefore one, three, two, six. The four gaps add to twelve, completing one circuit.

Starting at U's zero marker gives gaps one, two, three and six. They also add to twelve. Sorting either sequence from smallest to largest yields one, two, three, six. A record containing only that sorted list would discard the very ordering difference that prevents the match in this example.

A rigid rotation moves every marked site together while preserving the clockwise gap from each marker to the next. It cannot exchange the middle gaps independently. This explains why the equal-count and equal-sorted-gap checks were insufficient: they retained which lengths occurred but lost the sequence in which those lengths appeared around the circle.

A different starting marker rotates the gap list

The first gap written down depends on which marker begins the reading. Starting S's clockwise reading at marker one gives three, two, six, one. Starting at four gives two, six, one, three. Starting at six gives six, one, three, two. These are cyclic shifts of the same ordered gap sequence.

Accordingly, the comparison must allow a different starting marker without permitting arbitrary rearrangement. U's sequence one, two, three, six is none of S's four readings. In this example, the only gap of length one would have to line up with the other length-one gap; the next clockwise gaps would then disagree, three against two.

T's gaps, starting at its marker two, are one, three, two and six. They preserve S's sequence. Its successful offset was established directly by moving the four sites, so the gap observation also has a concrete matching construction behind it. Keep that offset rather than treating an unlabeled sketch's visual resemblance as the result.

Check whether a match is unique

Use a separate symmetric pattern E with markers zero, three, six and nine, and compare it with itself under the same twelve-step rule. Offsets zero, three, six and nine all reproduce E. A three-step turn changes which original marker occupies the top position, while leaving the set of marked sites unchanged.

This pattern has gaps three, three, three and three. Its repeated structure explains the multiple registrations. The comparison is compatible under the stated rule, but it does not identify one unique orientation. Save all four matching offsets if that distinction matters to the paper agreement. Do not import the original S-to-T result's unique offset into this separate example.

Distinguish nearby drawing questions

The mirrored paper template comparison keeps a locator on the required side of a named directed baseline. Here there is no signed coordinate side test or reflected candidate. The decision concerns clockwise gap order and allowed circular shifts of one fixed four-marker pattern.

The circle-and-compass drawing guide establishes a centre and radius before drawing. This registration model supplies its circle and twelve sites as inputs. It offers no compass adjustment, construction procedure, calibrated measurement or evidence that an actual canvas has the required dimensions or marked positions.

Keep the agreement with the result

Record the number of sites, numbering direction, source and target sets, fixed centre and radius, permitted offsets and matching results. Include whether a unique orientation is required. A note saying “four markers, gaps one, two, three, six” would leave out essential cyclic order in the S-to-U example.

Keep each offset calculation in a separate row. A row that takes zero-to-two from one turn and four-to-three from another would combine different operations. Check the same offset against every source marker, then compare the complete resulting set. The fit question concerns one rigid turn of the whole insert, not four independent placements.

Attach a payment enquiry to an actual order

If you separately consider an art-supply order on republeq.com and want to ask about USDT, identify the exact offer and intended quantity before discussing payment. The numbered sites and gap lengths in this exercise are fictional drawing inputs, not prices, cryptocurrency quantities or instructions to transfer funds. A successful paper registration establishes no payment option, package entitlement or compatibility of a real accessory. Read the choices and conditions actually presented for the intended transaction, and ask the responsible support contact to clarify essential information before deciding whether to proceed. If a cryptocurrency option is offered, establish the specified asset and network, requested amount, any stated validity period and the reference that would associate payment with the order. Do not infer a transfer destination from a canvas label, a circular arrow or an earlier transaction. Resolve the supplied contents and any actual fit requirement separately from the payment discussion. Ask about applicable delivery, cancellation, return and refund conditions, including the denomination of any approved refund and relevant deductions. Keep the answer with the offer it describes. Information about one assortment does not automatically describe a different package or quantity, and a changed order may require reviewing the resulting terms again. This article promises no present USDT availability, fee level, conversion result, confirmation time, refund amount or delivery outcome. You can check every permitted offset without purchasing either canvas offer. Preserve any eventual order acknowledgement and transaction record separately from the paper worksheet. One documents an actual transaction under its recorded conditions; the other records exact matches between stipulated marker sets. Neither a matching pattern nor a preferred payment asset fills a gap in the actual order information.

Retain order rather than only the gap collection

S matches T at offset two, while no permitted offset matches U. S and U have equal marker counts and the same sorted gap lengths, yet their clockwise gap orders differ. Preserve that order, allow only the stated cyclic starting changes and record the matching offsets. Those details explain the compatibility result under the complete fictional agreement.