Three-for-two grouping: keep the largest free item together

Keep the six required items fixed

Three-for-two grouping can change the amount payable even when the same six items are purchased. In this original fictional exercise, each group contains exactly three items and its lowest-value item is free. Putting the three largest values together produces a larger reduction than spreading them across the two groups. The difference follows from which items become the two group minima.

Use six symbolic items: A has value 12 tokens, B 11, C 10, D 3, E 2 and F 1. All six are already required. Their total before reductions is 39. These values are invented bookkeeping inputs, not prices of the padlocks linked below, cryptocurrency amounts or a statement that any seller offers this arrangement.

The rule is also invented. Divide these six items into exactly two groups of three, with every item used once. Remove the lowest value in each group from the total. No shipping, tax, fee, other discount, eligibility exception or purchase limit enters the calculation. We are choosing a partition of a fixed collection, without adding goods to obtain a larger headline reduction.

Read the native offers as separate accounts

The native Master Lock UTTK21489_1 padlock account describes a Proseries four-digit padlock with a 57mm die-cast zinc body. Its sole image shows a complete black Master ProSeries padlock and shackle. The pictured view does not verify the hidden mechanism, dimensions or material, and no security rating is established here.

The native Master Lock Proseries brass padlock account names a four-digit 57mm padlock. Its sole image shows a complete brass-colored Master ProSeries padlock with black trim. Color does not authenticate material. Neither captured account establishes the fictional promotion, six-item eligibility, checkout grouping or any of the token values used in this exercise.

Calculate the grouping with the larger free value

First put A, B and C in one group. Its values are 12, 11 and 10. C is the lowest, so its 10 tokens are removed. The remaining payment for that group is 12 plus 11, or 23. Calling C free means that the stipulated rule removes its value; it does not remove C from the collection of required items.

The other group contains D, E and F, with values 3, 2 and 1. F is lowest, so one token is removed. The remaining payment is 3 plus 2, or five. Across both groups, the reduction is 10 plus 1, giving 11, while the payment is 23 plus five, giving 28.

Check the total in both directions. The original 39 minus the reduction of 11 equals 28. Adding the four paid item values also gives 12 plus 11 plus 3 plus 2, or 28. These are two views of the same calculation. Keep the free item identities C and F beside the result so the reduction can be traced.

Compare a mixed grouping on the same inputs

Now make a separate partition: A, D and F in one group, and B, C and E in the other. The first group's values are 12, 3 and 1, so F is free. The second group's values are 11, 10 and 2, so E is free. The total reduction is one plus two, or three tokens.

The amount payable is therefore 39 minus three, giving 36. The direct payment check gives 12 plus 3 for the first group, and 11 plus 10 for the second. Fifteen plus twenty-one equals 36. The same required collection now costs eight more tokens than the first partition under the exact invented rule.

No item value changed between the two calculations. Neither grouping gained a delivery benefit or lost a needed item. The difference is that C becomes paid in the mixed grouping while E becomes free. Replacing a free value of 10 with a free value of two reduces the total saving by eight. F remains free in both partitions.

List the ten possibilities without double counting

A complete check can keep A in the first group and choose its two companions from the remaining five items. Once those companions are chosen, the other three items form the second group. Exchanging the names “first” and “second” would not create a different partition. This convention lists each unlabeled split once.

Start with companions involving B. ABC with DEF removes 10 and 1, totaling 11. ABD with CEF removes 3 and 1, totaling four. ABE with CDF removes 2 and 1, totaling three. ABF with CDE removes 1 and 2, also totaling three. Those four possibilities exhaust the choices that put both A and B together.

Continue with C but without B beside A. ACD with BEF removes 3 and 1, totaling four. ACE with BDF removes 2 and 1, totaling three. ACF with BDE removes 1 and 2, totaling three. The final possibilities are ADE with BCF, removing 2 and 1; ADF with BCE, removing 1 and 2; and AEF with BCD, removing 1 and 3.

Thus the last three totals are three, three and four. Across all ten partitions, one reduction is 11, three reductions are four and six reductions are three. Each payment follows by subtracting its reduction from the unchanged 39: 28, 35 or 36. This complete list confirms the first partition's result for these six particular values.

Prove why a larger reduction cannot occur

The table gives an exhaustive check, but a short bound explains the result. F has the smallest value of all six items. Whatever its two companions, F is the lowest in its own group. Its one token is therefore always one of the two removed amounts. Moving F cannot create a free value larger than one in that group.

The other group contains three of the remaining five items. Its lowest value cannot exceed C's value of 10. There are only two items larger than 10, namely A and B. A group of three whose minimum exceeded 10 would need three such items, but only two exist in this fixed collection.

The two free values can therefore total at most one plus ten, or 11. ABC with DEF reaches that bound because C is the minimum of ABC and F is the minimum of DEF. It follows that no allowed partition can require less than 39 minus 11, or 28 tokens. The bound and the construction establish the optimum together.

For these distinct values, attaining a minimum of 10 in the group without F requires that group to be ABC. D and E are both below 10 and would lower its minimum. The remaining group is then DEF. Reversing their order on the page changes neither membership nor payment, so it is the same unlabeled partition.

Check a changed value as a separate version

Suppose a new version changes C from 10 to nine while leaving every other input and the rule unchanged. The baseline total becomes 38. ABC still has C as its lowest item, now worth nine, and DEF still has F as its lowest, worth one. Their combined reduction is ten, leaving 28 payable.

The reduction has fallen by one, but the payment has stayed the same. This is consistent because the baseline also fell by one. A, B, D and E remain the paid items, and their values still sum to 28. Looking only at the smaller reduction would miss what happened to the original total.

The same bound adapts to this separately stated version. F still contributes one; the other group's minimum is at most nine because only A and B exceed nine. ABC with DEF reaches ten in total reduction. Save the changed input with its new baseline and bound, rather than placing its reduction beside the old baseline of 39.

Keep a grouping decision separate from other savings rules

The capped percentage discount comparison applies two rate-and-ceiling rules to the same subtotal. This exercise has no percentage or ceiling on a calculated discount. Its two removed amounts come from the lowest item in each selected triple, so the item membership of each group matters.

The minimum opening balance exercise checks cumulative payments and receipts in a stated order. Here no receipt arrives later and no intermediate funding path is tested. Our final 28 is a payable total under the invented grouping rule, rather than a result about available funds at every step.

Neither nearby calculation supplies a missing term for an actual promotion. A real offer might group items automatically, limit combinations or define eligibility differently. Establish its actual written conditions before applying any worksheet. The ten partitions here show what follows when grouping is permitted exactly as stipulated; they establish no permission at a particular checkout.

Do not change the need to improve the reduction

All six items were required before the comparison began. Removing F, adding another expensive item or buying a second copy would create a different collection and would require a new baseline. A larger reduction on that changed collection would not, by itself, prove a lower payment for the original need.

A useful record keeps six identities and values, the two group memberships, each minimum, the total removed and the amount still payable. Check that every identity appears once. Then recompute both the baseline-minus-reduction total and the sum of the paid values. A missing or duplicated item can otherwise make an attractive total answer a different question.

Keep an actual payment enquiry with its order

If you independently consider a padlock purchase on republeq.com and want to ask about USDT, begin with the exact offer, intended quantity and current order information. The tokens in this grouping exercise are invented arithmetic inputs, not cryptocurrency quantities, listed prices or payment instructions. The fictional free-item rule establishes no discount, combination right or available checkout method for either captured account. Read the options and conditions actually 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, the requested amount, any stated validity period and the reference that would associate payment with the order. Do not infer a transfer address or destination from a padlock image, a FREE card in the illustration or an earlier transaction. Ask separately about delivery, cancellation, return and refund conditions, including the denomination of any approved refund and applicable deductions. Preserve those answers with the offer they concern. A response about the zinc-described account does not establish the brass-described account's terms, and a change in quantity may require reviewing the resulting order information again. This article promises no present USDT availability, fee level, conversion result, payment confirmation, refund amount or delivery outcome. Its original partition calculation can be checked without ordering either padlock. Keep any eventual order acknowledgement and transaction record separately from the six symbolic values. Those records concern an actual transaction; the worksheet concerns the sum of two stipulated group minima. No arithmetic result here authenticates a merchant's payment instructions or establishes that a real basket qualifies for the invented promotion.

Save the partition that explains the payment

With A12, B11, C10, D3, E2 and F1 fixed, ABC with DEF removes 11 and leaves 28. The mixed ADF with BCE partition removes three and leaves 36. F must always be free, while the other group's free value cannot exceed ten. That bound explains why the first grouping attains the smallest possible payment under the complete fictional rule.