Recurring slot compatibility: keep the starting offsets

Recurring slot compatibility depends on where each list starts as well as how often it repeats. In this fictional worksheet, requirement A permits slots one, five, nine and so on, stepping by four. Requirement B permits three, nine, fifteen and so on, stepping by six. Their first shared slot is nine, followed by 21, 33 and later positions twelve apart.

The slots are nonnegative integers without a physical unit. Eligibility applies at one exact position, with no duration attached. These are invented requirements, not observed clock readings, calendar appointments or hardware signals. The exercise tests whether both conditions can hold together; it establishes no actual product fit or accessory interoperability.

Keep each start beside its step

Write requirement A as n equals one plus four times k, where k is a nonnegative integer. Starting with k equal to zero gives one. Increasing k by one gives five, then nine, thirteen, seventeen and 21. Every permitted A slot belongs to that same stated list.

Write B as n equals three plus six times j, with j also a nonnegative integer. Its first slots are three, nine, fifteen, 21, 27 and 33. The variables k and j count advances in their respective lists. They need not be equal when the two lists reach the same slot.

For example, A reaches nine after two advances from one, while B reaches nine after one advance from three. Thus k equals two and j equals one at this match. Requiring the two advance counts to agree would add a condition that the original compatibility question never supplied.

Check the first visible match

A's eligible positions below nine are one and five. B's only eligible position below nine is three. Those smaller candidates do not coincide. At nine, both conditions hold: nine equals one plus four times two, and nine equals three plus six times one.

This establishes the first shared nonnegative slot for the printed lists. Merely noticing that both contain nine would establish a match but would not, by itself, establish that it is the first. The check of all their earlier candidates supplies that additional conclusion.

Keep the original offsets in both substitutions. Replacing B's starting three with zero would create another list. A familiar repeat interval cannot justify changing the starting condition to make a calculation easier or a desired candidate appear eligible.

Use a complete twelve-slot block

Both eligibility patterns repeat after adding twelve. For A, twelve is three advances of four. For B, it is two advances of six. Adding twelve to a shared slot therefore preserves both requirements. Starting from nine produces 21, 33, 45 and 57.

To check whether those are all the shared slots, inspect the residues from zero through eleven. A's condition permits residues one, five and nine in this block. B's permits three and nine. Their only common residue is nine. Do not omit zero from the inspected block merely because neither list starts there.

Every nonnegative integer can be written as twelve times a nonnegative integer t plus one residue from zero through eleven. The eligibility patterns just checked repeat in each such block. A shared slot must therefore have residue nine, giving n equal to nine plus twelve times t.

Conversely, each number of that form satisfies both lists. Nine plus twelve t equals one plus four multiplied by (two plus three t), and it equals three plus six multiplied by (one plus two t). The advance counts two plus three t and one plus two t remain nonnegative for every allowed t.

Thus the complete shared family is nine, 21, 33 and every subsequent nine-plus-twelve-t position. The argument checks both directions: every common slot must be in the family, and every member of the family is permitted. The full-block argument checks the positions that a short list of successful guesses leaves unexamined.

Separate the repeat gap from compatibility

The number twelve describes the spacing between shared positions in this example. It does not, on its own, establish that two lists have any shared position. Their offsets determined the residue intersection. Keep that distinction visible before using a common repeat interval as a compatibility answer.

Make a separate version that retains A but changes B's start from three to two. The new B list is two, eight, fourteen, twenty, 26 and so on. Its step remains six. A still starts at one and steps by four. The pair of step sizes has not changed.

Every A slot is odd, because adding a multiple of four to one preserves oddness. Every new B slot is even, because adding a multiple of six to two preserves evenness. No integer slot can be both. These two lists therefore have no shared eligible position.

Waiting another twelve slots cannot remove that conflict. Each list repeats its own parity pattern after twelve, and the mismatch remains. An answer saying only “they repeat every twelve” would leave the central compatibility question unanswered for this changed-offset version.

Change an offset as a new input

For another separate version, let B start at five and still step by six. Its slots are five, eleven, seventeen, 23, 29 and 35. A remains one, five, nine, thirteen, seventeen and 21. This time their first common position is five, not nine.

In the full twelve-residue block, the changed B condition permits five and eleven. A still permits one, five and nine. Their intersection is now five, so the complete shared family becomes five plus twelve times t for nonnegative integer t. The repeat gap stays twelve while the first match changes.

Preserve all three versions separately. B starting at three gives first match nine; starting at two gives no match; starting at five gives first match five. Each uses the same step sizes. Transferring a first-match result between these records would silently drop the input responsible for their difference.

Add a lower bound without restarting the lists

Return to the original starts one and three. Suppose a new request asks for a shared slot at or after ten. The lists themselves do not restart at ten. Their shared family remains nine plus twelve t, and the first member meeting the added lower bound is 21.

Nine is a genuine shared slot but is below the requested bound. Twenty-one satisfies the bound and both original conditions. If an inclusive upper limit of twenty were also imposed, no shared slot would remain in that permitted range. An upper limit of 21 would include the match at 21.

Write whether an endpoint is included. A request strictly after 21 excludes 21 and makes 33 the next shared position. These changes concern the queried range, not the list offsets. Keep the range and the recurring requirements in different fields so another reader can reproduce the same question.

Retain the exact eligibility model

The worksheet treats eligibility as a point at an integer slot. A real activity that remains available for several slots would require a duration or interval rule. Two intervals can overlap even when their starting positions differ. That is another model, and these point-membership results do not settle it.

Likewise, a rule allowing only a finite number of repeats would need an explicit stopping point. The complete families above assume that each given list continues indefinitely. If a list ends before a proposed shared position, the printed infinite-list conclusion cannot establish that the finite version permits it.

None of the slot numbers is an actual time, date, time zone or clock setting. The offsets and steps are supplied arithmetic inputs. A physical synchronization question would need its own verified devices, settings and observations; the paper model supplies no such evidence.

Read the actual clock accounts separately

The Seiko Time Tutor clock account names QXA793RLH. Four images show the complete red-rim clock, side, back and room context, with Seiko labeling. Native size twelve and dimension wording remain unmeasured declarations. Quiet motion, accuracy, battery life, authenticity and actual displayed time are not verified here.

The Bulova Annette II clock account names B1929. Its sole image shows a complete dark reddish-brown arched mantel clock with Bulova wording and Roman numerals. No measured dimension, material, chime, movement performance or authenticity is established. The photograph does not inspect a delivered parcel.

Neither selected account supplies the fictional slot lists. Room props are not verified supplied contents, and pictured dials provide no rule for this exercise. The clock references offer optional context without establishing synchronization, accessory fit or a demonstrated pairing.

Keep nearby compatibility questions distinct

The relative offset compatibility guide checks exact signed differences around a loop and distinguishes a compatible family from an absolute anchor. Here each condition selects members of a recurring list. A starting offset and repeat step together determine which slots belong.

The last digit power-sum guide uses an exponent's position within a repeating last-digit cycle. Our index is an eligibility slot, not an exponent. That arithmetic pattern supplies neither the starts nor the shared-slot result needed here.

Ask about an actual order separately

If you separately consider a clock order on republeq.com and want to ask about USDT, begin with the exact offer and intended quantity. The fictional slots, offsets and repeat steps in this worksheet are not prices, cryptocurrency amounts or payment instructions. Correct arithmetic establishes no checkout method or actual order status. Read the payment options and conditions presented for the intended transaction, and ask the responsible support contact to clarify essential missing 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 associating payment with the order. Do not infer a transfer destination from a product image, worksheet heading or clock face. Resolve material product questions separately, including actual supplied contents and any intended use you need the item to support. Ask about applicable delivery, cancellation, return and refund conditions, including the denomination of any approved refund and relevant deductions. Keep each answer with the offer it concerns. Information about one account does not automatically describe another quantity or purchase, 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 inspect every fictional recurring list without purchasing anything. Preserve any eventual order acknowledgement and transaction record separately from the worksheet. Those records concern an actual transaction under its documented conditions; the calculations test explicitly stated eligibility requirements. A preferred payment asset or a plausible shared slot cannot fill a gap in actual order information.

Save the starts with the result

Recurring slot compatibility for the original lists gives first match nine and complete family nine plus twelve t. The changed start of two is incompatible, while a start of five gives another first match. Retain both starts, both steps and the permitted domain beside any common-gap calculation.