Pairwise count totals: recover all three bags

Pairwise count totals can recover three individual counts when every pair is recorded exactly. In this original paper problem, bags A and B together contain 17 counters, A and C contain 19, and B and C contain 22. The individual counts are seven, ten and twelve. The useful check is why those numbers are forced and whether they satisfy the declared whole-count rules.

All bags, counters and observations here are fictional. Each bag contains a nonnegative integer number of counters. The three bags are distinct, and no counter belongs to more than one bag. Every recorded pair includes all counters in its two named bags. These assumptions define the fictional problem.

Write the three pair records separately

Let a, b and c denote the counts in bags A, B and C. The supplied equations are a plus b equals 17, a plus c equals 19, and b plus c equals 22. Keep each pair label beside its number. A bare list of 17, 19 and 22 would lose which two bags contribute to each total.

The records describe the same three bags without any counter being added or removed between observations. If a different version changed a bag's contents, its equations would need a separate record. Combining one pair from an earlier version with two from a later version would not preserve the fixed-count problem stated here.

Add the pairs and count each bag twice

Add the three recorded totals: 17 plus 19 plus 22 equals 58. In that addition, A appears in the first and second pairs, B in the first and third, and C in the second and third. Every bag's count is therefore included exactly twice.

The sum of the pair totals is twice the combined count across all three bags. Dividing 58 by two gives 29. This is the count of counters in the three distinct bags together, under the stated assumptions. It is not another pair total or an average count per bag.

Writing the labels explains the division. The left side is a plus b plus a plus c plus b plus c. Regrouping gives two lots of a, two of b and two of c. This particular overlap is the reason for dividing by two.

Subtract the pair that leaves each bag out

The combined count of 29 includes all three bags. The recorded B-and-C pair contributes 22 of those counters. Subtracting 22 from 29 leaves seven for A. Similarly, removing the A-and-C total of 19 leaves ten for B. Removing the A-and-B total of 17 leaves twelve for C.

Keep the omitted bag visible in each subtraction. A equals the combined count minus the B-and-C total. Subtracting A-and-B instead would recover C, not A. A correct subtraction with the wrong bag label would still give an incorrect reconstruction record.

Now test all three original equations. Seven plus ten equals 17; seven plus twelve equals 19; ten plus twelve equals 22. The reconstructed counts are nonnegative integers, and their combined count is 29. Thus they meet both the pair equations and the whole-count conditions of this version.

Explain why another assignment cannot work

Any assignment satisfying the same three pairs must also have twice its combined count equal to 58. Its combined count must therefore be 29. The B-and-C pair remains 22, forcing A to be seven. The other two subtractions likewise force B to be ten and C to be twelve.

This proves uniqueness for the defined equations instead of merely finding one attractive answer. Guessing seven, ten and twelve and checking them would establish that they work. The combined-total argument additionally establishes that every working assignment has those same three counts.

The argument depends on having every pair for exactly three bags. It does not show that any three pair records from a larger collection determine all its individual counts. Preserve the bag set and pair membership rather than transferring this conclusion to a different recording pattern.

Keep a reusable formula with its conditions

Write p for the A-and-B total, q for A-and-C, and r for B-and-C. Adding them gives twice the combined count, so that count is p plus q plus r, divided by two. Subtracting r then gives a equal to p plus q minus r, divided by two.

The corresponding formulas are b equal to p plus r minus q, divided by two, and c equal to q plus r minus p, divided by two. Each numerator adds the two pairs containing the desired bag and removes the pair containing only the other bags.

These formulas give one candidate over ordinary real numbers for any stated real pair totals. Whether the candidate is a permitted whole count remains another check. The original problem requires nonnegative integers; a fraction or negative value cannot be accepted merely because it satisfies the equations algebraically.

An odd pair-total sum cannot give whole counts

Make a separate version with pair totals 17, 19 and 23, leaving the bag labels in the same order. Their sum is 59. If whole counts satisfied these pairs, twice their whole-number combined count would equal 59. Twice an integer is even, so this version cannot have a whole-count solution.

The formulas still produce a real-number assignment: A is 6.5, B is 10.5 and C is 12.5. Those values satisfy the three pair equations, including 10.5 plus 12.5 equals 23. They fail the stipulated integer condition. Rounding them would change the pair totals and would not repair the original record.

A problem that explicitly allowed continuous quantities could have a different permitted domain. This one counts whole counters. Keep that rule beside the answer rather than silently replacing counters with fractional amounts just to make a candidate acceptable.

An even total can still leave a negative count

Now consider pair totals four, five and eleven in the same pair order. Their sum is 20, giving a combined count of ten. The formulas produce A equal to minus one, B equal to five and C equal to six. All three values are integers, but A violates the nonnegative-count condition.

Checking only that the combined total was whole would therefore miss a real failure. Here the A-and-B and A-and-C totals add to nine, less than the B-and-C total of eleven. Their difference is twice A, so a negative A follows directly from the given pairs.

The arithmetic identifies incompatibility with the declared count rules without identifying which record was mistaken. A real uncertain observation would need its own evidence. Do not change eleven to a convenient smaller number and report that the original problem has been solved.

Allow zero at the boundary

A separate version with pair totals four, five and nine gives a combined count of nine. A is zero, B is four and C is five. Each pair checks, and every count is a nonnegative integer. An empty bag is permitted by the stated model.

For nonnegative integer pair totals, two conditions fully characterize whole-count feasibility here. Their sum must be even, and no one pair total may exceed the sum of the other two. The second condition makes all three reconstruction numerators nonnegative. The first makes each numerator even, since subtracting twice a pair total preserves parity.

Those conditions are sufficient as well as necessary. The formulas then produce nonnegative integers, and adding the reconstructed A and B cancels the opposite terms to recover p. The other two additions recover q and r. This verifies the whole list rather than relying on a plausible combined total alone.

Leave a missing pair unresolved

If only A-and-B equals 17 and A-and-C equals 19 are supplied, the third equation is missing. Let A be an integer t. Then B must be 17 minus t and C must be 19 minus t. Every integer t from zero through 17 gives a nonnegative assignment satisfying those two recorded pairs.

There are eighteen such assignments. Choosing seven happens to reproduce the original complete example, but the two remaining equations do not force seven. The missing B-and-C total would distinguish the possibilities. Preserve that missing input rather than borrowing 22 from another version of the worksheet.

Keep the actual ruler accounts separate

The ANYWAY.GO twelve-centimeter ruler account retains its catalog label. Its sole image shows a complete grey ruled blade with integral handle and hole, marked TISUR. Visible-gauges and total-length wording are native declarations, not measurements here. No maker relationship, material, strength, calibration or authenticity is established.

The Anyway.go eighteen-inch ten-pack account retains those native terms. Its sole image has NOKKO ten-pack packaging and long front/back graduated-rule views, with lower ends cropped. The array does not independently verify ten delivered rulers or their full lengths. No maker equivalence, material or measurement accuracy is established.

Neither selected account supplies the fictional bag counts. Catalog labels and pictured maker names remain distinct observations, without an alias claim. The paper problem uses no ruler measurement and requires no purchase.

Keep neighboring arithmetic questions distinct

The relative offset compatibility guide checks signed differences around a loop and distinguishes a compatible family from an absolute anchor. Here the supplied relations are pair sums, which force one candidate before its nonnegative whole-count conditions are tested.

The pooled ranking reversal guide compares group fractions under different mixtures and an explicitly shared mix. This problem reconstructs counts from fixed pair sums. It supplies no completion percentages, ranking of alternatives or group weights to compare.

Keep a purchase enquiry with its actual order

If you separately consider a ruler order on republeq.com and want to ask about USDT, begin with the exact offer and intended quantity. The fictional bags, counters and pair totals 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 ruler marking. 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 pair total 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 recover expressly stipulated counts. A preferred payment asset or a plausible count cannot fill a gap in actual order information.

Save the recovered counts with the original pairs

Pairwise count totals of 17, 19 and 22 force the individual counts seven, ten and twelve under the stated bag labels. Preserve the equations, domain and complete substitution check together. An odd total, a negative candidate and a missing pair leave different issues, each requiring its own truthful record.