Understand chains of near matches

A connecting chain can leave a pair outside the rule

In this original paper example, a value of 10 is near 11, and 11 is near 12, but 10 is not near 12. There is no contradiction once “near” has a precise definition. Chains of near matches answer a different question from a requirement that every pair within a group must be near. Before accepting a group, state which of those two conditions it needs to satisfy.

Our rule uses invented numerical labels, not measurements of actual bookends. Take five paper labels: A has value 10, B has 11, C has 12, D has 20 and E has 21. Call two distinct labels near when the larger value minus the smaller is no more than one. Exactly one qualifies; a gap larger than one does not. The rule is deliberately stipulated for this exercise.

Nothing here establishes a visual similarity test, a product tolerance, a material grade or a method for deciding whether two real objects are interchangeable. The values do not describe the dimensions, weight or color of either linked offer. Paper is sufficient to check the example. Keep the numerical rule beside the five labels so a familiar word such as “near” cannot quietly acquire another meaning halfway through the comparison.

Check direct pairs before drawing groups

Begin with A and B. Their values are 10 and 11, so the gap is one and the pair qualifies. B and C also qualify because 11 and 12 differ by one. A and C do not qualify: their values differ by two. These three checks already establish the central distinction. Two qualifying pairs can share a middle label without making their outer pair qualify.

Give D and E the same direct check. Values 20 and 21 differ by one, so that pair is near. No label in the first three is near either of these last two. The smallest cross-group gap is between C at 12 and D at 20, which is eight. The other cross-group gaps are larger, and therefore also fail the stated one-unit limit.

For a complete audit, record the ten distinct pairs without treating a reversal as another relationship. AB has gap 1; AC, 2; AD, 10; AE, 11; BC, 1; BD, 9; BE, 10; CD, 8; CE, 9; and DE, 1. Only AB, BC and DE pass. This list checks the chosen relation for the five stated values; it is not a claim about any real collection.

Draw a short line for each passing pair and leave failed pairs unconnected. A connects to B, B connects to C, and D connects to E. A line records a direct passing comparison. It does not mean that the endpoints look alike or share a manufacturer, and a longer drawn line would not change its numerical meaning. The values and the rule supply the evidence, rather than the distance between cards on the desk.

Define what a chain is allowed to do

For the first kind of group, allow a route through one or more passing pairs. A label belongs with another if a chain of those direct links connects them. Under that definition, A and C belong to the same linked group because the route A to B to C uses two passing comparisons. A and C need no direct line for this particular condition.

The linked groups in our example are therefore A, B, C together and D, E together. A route can reach B or C from A, but none of the recorded links leads from that group to D or E. Within the second group, D and E have a direct connection. Retain the actual links beside the membership list so a reader can distinguish a direct comparison from a route through a middle label.

Saying “A and C are connected through B” is accurate. Saying “A and C pass the one-unit comparison” is not. The route has not changed their values or reduced their gap of two. A membership rule can permit an indirect connection without granting every possible direct comparison inside the resulting group. Use wording that preserves that distinction when reporting the result.

You can inspect the chain one step at a time. First verify AB using the original values, then verify BC using those original values. Do not replace B with a calculated intermediate value or apply the gap operation twice to produce a new output. This exercise tests a yes-or-no relationship between fixed labels. Its route does not calculate a third value for the next comparison.

Require every pair when that is the actual condition

Now change the group requirement, while keeping all five values and the one-unit rule unchanged. Require every distinct pair inside a proposed group to pass. The proposed group A, B, C fails because AC has gap two. The fact that its other two pairs pass does not satisfy the missing comparison. Identify the exact failing pair rather than describing the whole group as nearly acceptable.

The two-label groups A, B and B, C each pass this every-pair condition, as does D, E. Each contains only one distinct pair to check. A three-label group needs its third pair checked as well as the two links that could draw a chain. In this example, that extra check is exactly where the three-label proposal fails.

There is an additional choice if every label must be placed into a group. If overlapping groups are allowed, both A, B and B, C can be retained, with B appearing in two groups. If each label must appear in only one group, those two groups cannot both be selected unchanged. The requirement about overlap is separate from the numerical closeness test.

Two possible disjoint arrangements are A, B together with C alone and D, E together; or A alone with B, C together and D, E together. In either arrangement, every non-singleton group passes its internal pair test. This example does not select one arrangement as best. A further preference would need to be stated rather than inferred from the fact that both arrangements meet the current condition.

Keep the linked-group result separate from these every-pair arrangements. The linked rule gave a three-label group A, B, C without choosing between AB and BC. The every-pair rule rejects that three-label group and leaves a choice about grouping or overlap. Writing only “group similar labels” would hide the decision that makes those different answers appropriate.

Add a bridge without changing the endpoints

Set aside the five-label worksheet for a moment and begin a smaller version containing only A at 10 and C at 12. The pair fails because its gap is two. Now add B at 11, retaining both original values. AB passes and BC passes, so the added label creates the route A to B to C. AC still fails the direct comparison.

This bridge example explains how linked membership can change when a label is added even though an existing pair's values remain unchanged. The new route is evidence of a changed collection of available links. It is not evidence that the earlier AC calculation was wrong. Preserve the before-and-after label sets so another reader can see which input changed.

Removing B from the three-label chain has the reverse effect in this particular example: it removes the only route joining A and C. Their values stay 10 and 12 and their direct gap stays two. The conclusion concerns the links actually present here. It does not assert that removing any middle-looking label from any larger diagram will disconnect its endpoints; a different diagram could contain another route.

A changed threshold is a changed rule

Return to the original five labels and make a separate worksheet with a limit of two instead of one. AC now passes because its gap is exactly two. In that revised worksheet, all three pairs among A, B and C pass. Their values have not moved closer; the condition used to accept a pair has changed. Label the revision so its result cannot be mistaken for the earlier one-unit test.

Likewise, requiring a gap strictly less than one would differ from the original “no more than one” wording. The original passing pairs all have gap exactly one, so each would fail that stricter rule. These comparisons show why the boundary word matters for our stipulated data. They supply no recommended tolerance for an actual product, material or measurement process.

Keep related paper questions in their own scope

The gap-rule grouping FAQ calculates a new numerical output at each stage and compares two choices of first pair. This article leaves every label's value fixed and tests direct relationships between them. Following a chain is not a nested gap calculation, and no intermediate numerical result replaces B.

The guide to choosing display pairs distinguishes an unordered selection from two named left/right positions. Here pair identities simply organize the closeness audit. The question is whether each chosen pair passes and what that implies for a group, rather than whether reversing two display positions counts as another arrangement.

The actual bookend offers supply only context

The native NOBRAND mountain bookend offer identifies a set of two. Its single gallery image shows a gold-and-bronze-colored pair with mountain reliefs and supporting bases. The catalog brand field is retained without using the visible side marking to authenticate a manufacturer. The image does not establish load capacity, composition or a physical one-unit matching tolerance.

The native Theia Stones ABAX-0090 geode bookend offer identifies a pair and explicitly says its stock photos do not show the exact set a buyer will receive. Its three gallery images show different pale and dark examples with polished faces and rough outer edges. Those examples are not all supplied pairs, and they establish no exact delivered appearance, material authentication or protective performance.

Separate any purchasing question from the relation test

If you separately investigate either bookend offer on republeq.com and would like to ask about USDT, use the actual order information rather than the paper labels as your starting point. The five invented values are not product prices, payment amounts or measurements that select an order. This article does not establish that a particular seller, checkout, account or location currently offers cryptocurrency payment. Read the methods and terms presented for the exact transaction you are considering. If a cryptocurrency option is offered, establish the specified asset and network, the payable amount, any stated validity period and the way the payment would be connected with your order. Resolve unclear details with the responsible support contact before deciding whether to act. A stock photograph, a matching brand name or a route through the paper diagram cannot authenticate a payment destination. Do not infer transfer instructions from any of them. Ask about the order's cancellation and refund conditions as well, including the currency of any return and the requirements that would apply. Keep the answers alongside the product reference and terms they concern. The geode offer's warning about variable stock appearance remains relevant to that offer, while the mountain offer retains its own identity; a payment answer about one does not silently cover the other. You can complete every comparison in this article without buying either pair. No present USDT eligibility, discount, fee level, conversion result, refund amount, dispatch date or successful delivery is promised here. A later purchase record would describe an actual transaction, while this worksheet records only the declared values, direct comparisons and chosen group requirement.

State which kind of group passed

The original one-unit rule gives direct links AB, BC and DE. It connects A, B, C through a chain, yet AC remains a failing pair. If your requirement is connection by a route, retain that chain and name its links. If the requirement is that every pair be near, check AC and reject the three-label proposal. Keep the threshold, label set and overlap condition with the result so the word “group” cannot conceal which question was answered.