Check relative offset compatibility around a closed loop

A closed loop can satisfy relative requirements

Relative offset compatibility depends on what the arrows require, not simply on whether they form a loop. In this original paper exercise, three labels have exact integer values. The requirements say B equals A plus 2, C equals B plus 3, and A equals C minus 5. All three can hold together, even though following the arrows brings you back to A.

Use A equal to 10, B equal to 12 and C equal to 15 as a first witness. The first change takes 10 to 12, the second takes 12 to 15, and the final change takes 15 back to 10. The closing requirement returns to the same value assigned to the same label. We do not need to delete an arrow to make this version consistent.

These are invented numerical labels, not measured positions, product dimensions, electrical connections or assembly instructions. A, B and C identify three entries in a paper model. The signed changes are the complete requirements of that model. No photographed paper clip supplies a distance, a coordinate, a tolerance or a rule about how actual parts connect.

Write each direction beside its signed change

Read the first requirement as a change from A to B. Starting with the value of A, add 2 to obtain B. Read the second from B to C and add 3. Read the last from C to A and subtract 5. The source label and destination label matter just as much as the written number.

A minus sign here specifies a decrease in the numerical label while following that arrow. It does not say that a physical component moves left, down or backward. There is no physical coordinate system in the exercise. Likewise, the plus sign supplies no statement that one task happens later than another task.

Keep the three requirements written as a single versioned list: B = A + 2; C = B + 3; A = C - 5. If a copied diagram reverses an arrow, check its sign against this list before using the picture. A decorative arrow without a defined direction would leave the numerical requirement unclear.

Check the return to the same starting label

Follow the full route from an initial A. After the first two changes, the label at C is A plus 5. Applying the final minus 5 brings the result back to A. The total change around this route is 2 plus 3 minus 5, which equals zero. That is the closure check for these three stated requirements.

The zero total is not merely a pleasing pattern in the numbers. Returning to A means returning to the value already assigned to A. If the route produced another value, the record would demand two different exact values for that same entry. Here it produces the original value, so the closing line agrees with the first two.

For this complete three-line model, we can also construct an answer directly. Choose A as 10, calculate B as 12 and C as 15, then check that 15 minus 5 equals 10. Every requirement has been checked against the same three values. There are no additional restrictions hidden in the exercise.

Separate a compatible family from an absolute anchor

The witness 10, 12, 15 is not the only answer. A equal to zero gives B equal to 2 and C equal to 5, with 5 minus 5 returning to zero. A equal to 27 gives B equal to 29 and C equal to 32, with 32 minus 5 returning to 27.

For any chosen nonnegative integer t, the assignments A = t, B = t + 2 and C = t + 5 satisfy this version. Adding the same amount to all three values preserves every required difference. The requirements therefore determine their relative offsets without selecting one absolute starting label.

If the worksheet additionally stipulates A = 10, that anchor selects B = 12 and C = 15 from the family. Keep the anchor in its own field. A statement that the relative requirements are compatible is different from a statement that the record has supplied a particular absolute value.

Do not choose 10 because it appears on a cover and then report that the original requirements forced it. The illustration shows one witness. The written model supplies the differences. An anchor becomes part of the model only when it is explicitly added as an input, not when a reader happens to prefer a convenient example.

Keep the inconsistent alternative as a separate version

Now make a separate copy whose last requirement is A = C - 4. Retain B = A + 2 and C = B + 3. Starting with A = 10 still gives B = 12 and C = 15, but the closing line now produces 11. It does not return to the value 10 assigned to A.

The net change around this alternative route is 2 plus 3 minus 4, which equals 1. Substituting through the list demands A = A + 1. No exact integer can satisfy that equation. The failure belongs to the whole simultaneous list, rather than to the particular starting value chosen for the demonstration.

Changing the starting label does not remove that residual. If A starts at 27, the first two lines produce 29 and 32, and the alternative closing line produces 28. The route again finishes one above its starting label. A new anchor moves the values while preserving the same disagreement.

Preserve the original compatible version and this inconsistent alternative separately. A note saying “changed closing offset from minus 5 to minus 4” identifies the changed input. Calling both diagrams the same model would hide the exact difference that makes one close and the other fail.

A passing pair does not settle the full alternative

The first two requirements in the inconsistent alternative can coexist: A = 10, B = 12 and C = 15 satisfy them. The failure appears when the closing requirement is checked too. Retaining a plausible first pair while ignoring the last line would answer a smaller question than the one printed.

The second and third alternative requirements can also coexist by themselves. Set B = 12 and C = 15, then set A = 11 so that A equals C minus 4. That assignment satisfies those two lines, but it fails B = A + 2 because 11 plus 2 is 13, not 12.

Likewise, A = 10, B = 12 and C = 14 satisfy the first line and the alternative closing line. They fail C = B + 3, which would require 15. These examples explain why separate two-line successes do not establish one shared assignment satisfying all three lines at once.

Reverse a direction only with its corresponding sign

In the compatible version, B = A + 2 can be rewritten as A = B - 2. C = B + 3 can be rewritten as B = C - 3. A = C - 5 can be rewritten as C = A + 5. Each rewrite describes the same required difference while reversing the direction used to read it.

Following the compatible loop backward therefore uses changes minus 2, minus 3 and plus 5. They still add to zero. Merely reversing an arrow while keeping its old sign would create a different requirement. Treat a direction-and-sign pair as one piece of the record when copying or reviewing the worksheet.

This exercise uses exact integer arithmetic without a clock face or modular wrap. A net change of 1 is a discrepancy here, not a return to the same label after a cycle. It also supplies no tolerance within which two different numbers may be treated as equal. Those would be additional rules requiring a separately defined model.

Do not turn a consistency result into a repair instruction

If the inconsistent alternative represented somebody's intended requirements, the arithmetic would identify a conflict without identifying which line was mistaken. Changing the last offset to minus 5 would create a compatible list, but it would not prove that this was the intended correction. Another requirement might have been copied incorrectly instead.

Keep the governing input and the proposed change separate. In a real application, obtain the applicable requirement rather than choosing whichever edit produces a convenient zero total. Even the compatible paper model would not establish a real part's fit, an accessory's suitability or the accuracy of any measurement.

Distinguish other kinds of compatibility checks

The circular instruction schedule uses strict before-relations between once-only tasks. Its loop demands an impossible ordering. Our signed differences permit a decrease on the closing arrow, so all three requirements can remain when their net change is zero. A loop's meaning determines what must be checked.

The guide to pairwise closeness distinguishes neighboring links from an all-pairs condition. Here each directed arrow states an exact numerical difference, and the question is whether one assignment satisfies the entire list. No tolerance chain or clustering rule supplies these offsets.

Keep the native paper-clip accounts limited

The native Promarx multicolor paper-clip account describes 200 clips per pack and a case of 48. Its sole gallery image shows a branded blister pack. That image does not authenticate a delivered case count or establish retention performance.

The native Baumgartens Plastiklips account describes medium no. 4 paper clips. Its five images show clips, packaging and notebook-use context. The notebook is a photographic prop. Neither account supplies the numerical offsets or proves material, nonmarking or fit claims for this worksheet.

Keep any USDT enquiry with the actual order

If you independently consider a paper-clip order on republeq.com and want to ask about USDT, use the exact item and quantity as the reference for that enquiry. The labels 10, 12 and 15 in this exercise are fictional numerical assignments, not product prices, cryptocurrency amounts or payment instructions. This article does not establish that a particular payment option is currently offered for an account, location or basket. Read the choices and conditions actually presented for the intended transaction, and obtain clarification from the responsible support contact where an essential detail is missing. If a cryptocurrency option is offered, establish the specified asset and network, the payable amount, any stated validity period and the reference associating payment with that order. A correctly drawn loop cannot identify a transfer destination or confirm that a transaction has been credited. Do not infer instructions from a product photograph, an earlier order or a diagram's arrow direction. Ask separately about applicable delivery, cancellation, return and refund conditions, including the denomination and deductions that would govern an approved refund. Keep those answers attached to the offer they concern. An answer about the Promarx account does not automatically describe the Baumgartens account, and a changed quantity may require checking the resulting order information again. No present USDT eligibility, discount, fee level, conversion result, refund amount, processing time or delivery outcome is promised here. You can test every stated difference on paper without ordering either product. Preserve any eventual purchase acknowledgement separately from the worksheet. One record concerns the actual transaction and its documented terms; the other records which exact requirements and label assignments were checked. Neither record supplies evidence missing from the other.

Retain the requirements and the witness together

The compatible version keeps changes plus 2, plus 3 and minus 5, with net change zero and witness 10, 12, 15. The alternative closing change of minus 4 creates a residual of 1 and no shared exact assignment. Save the direction, sign, version and any anchor together. A closed loop can remain compatible; the actual requirements determine whether this one does.