Build a six-participant pairing plan and check every pair

A six-participant pairing plan can look complete while still containing a hidden mistake. To verify it, ask two separate questions: does every participant appear exactly once in each round, and does every possible pair appear exactly once across the whole plan? This tutorial builds a fictional paper example, checks both requirements, deliberately introduces an error, and repairs it.

The participants are six invented identities labelled A, B, C, D, E and F. The worksheet contains five rounds, with three unordered pairs in each round. Here, a round is simply a group of written relationships. The example supplies no dates, durations, scoring system, official competition rules or instructions for physical gameplay. Its purpose is to demonstrate construction and verification.

Define the identities and the meaning of a pair

Start by writing the complete participant list: A, B, C, D, E and F. These labels remain fixed throughout the exercise. Changing a label midway would change the identity being tracked and make earlier audit marks unreliable. Leave enough space below the list for five separate rows and a second area containing the required pairs.

A pair contains two different participants. Because the pairs are unordered, AB and BA name the same relationship. Neither spelling assigns a first position, a preferred role or any other status. Write the letters alphabetically within each pair so that the same relationship always has the same name. For example, use DF rather than alternating between DF and FD.

The target is that each participant pairs with each of the other five exactly once. A must therefore appear with B, C, D, E and F somewhere in the completed worksheet. The same requirement applies to every other label. Checking A alone cannot establish that the remaining relationships are complete, even if A has five different partners.

Build the round-robin pair checklist

Prepare the round-robin pair checklist before arranging the rounds. Begin with pairs involving A: AB, AC, AD, AE and AF. Then continue with B, adding only relationships not already listed: BC, BD, BE and BF. Do not add BA, because AB already represents that pair. This method gives each possible relationship one place on the checklist.

Continue with CD, CE and CF, followed by DE and DF, and finally EF. The complete checklist is AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF and EF. There are five entries beginning with A, four with B, three with C, two with D and one with E.

Adding those groups gives 5 + 4 + 3 + 2 + 1 = 15 unique pairs. Another way to understand the total is that six participants each have five possible partners, giving thirty participant-to-partner references. Each unordered pair appears in two such references, once from each participant’s viewpoint. Dividing thirty by two therefore gives the same fifteen relationships.

Work out how many rounds the worksheet needs

If all six labels appear exactly once in a round, that round contains three pairs. Each pair uses two labels, so three pairs account for the six available identities. Four pairs would require additional appearances, while two pairs would leave two participants unassigned. This is a property of the stated paper exercise, not a claim about any external activity.

The complete checklist contains fifteen required pairs, and each full round contributes three pair positions. Five rounds therefore provide fifteen positions. The individual-participant count agrees: each label needs five different partners and receives one partner per round, so five rounds provide the necessary five appearances. These counts establish the required size of this example, but not whether its contents are correct.

Keep the distinction between positions and identities clear. Fifteen written pair positions could contain fifteen different relationships, or they could contain duplicates that displace required relationships. Similarly, six written label positions within a row could include a repeated participant. Counting positions is a useful preliminary check; inspecting which identities occupy them is the substantive audit.

Write the exact five-round example

Use the following first two rows. Round one is AF BE CD. Round two is AE DF BC. In the first row, A pairs with F, B pairs with E, and C pairs with D. In the second, A pairs with E, D pairs with F, and B pairs with C. Keep each round on its own worksheet line.

Continue with round three as AD CE BF and round four as AC BD EF. Complete the example with round five as AB CF DE. The five rows are therefore AF BE CD; AE DF BC; AD CE BF; AC BD EF; AB CF DE. Preserve these exact pair identities when copying the example into your own notes.

Notice that A’s successive partners are F, E, D, C and B. This provides a convenient starting structure, but it does not independently validate the other two pairs in each round. Completing the remaining spaces requires attention to relationships among B through F. The audits that follow establish whether the entire arrangement satisfies the stated requirements.

Run the first audit within each round

The first audit checks participant appearances one round at a time. Start with AF BE CD. AF accounts for A and F, BE accounts for B and E, and CD accounts for C and D. Compare those six appearances with the original participant list. Every label from A through F appears once, with no repetition or omission.

Apply the same procedure to round two, AE DF BC, and round three, AD CE BF. Each row contains A, B, C, D, E and F exactly once. Then inspect round four, AC BD EF, and round five, AB CF DE. Both also contain the complete six-label set without duplication. All five rounds pass the first audit.

Do not replace this comparison with a simple count of six letters. A hypothetical row AF AB CD also contains six written letter positions, yet A appears twice and E is missing. Each individual pair has two different labels, but the row still fails. The first audit must examine the combined labels across all three pairs, not merely inspect pairs separately.

Use the checklist to check repeated pairings

The second audit examines relationships across the complete plan. Start with a clean fifteen-entry checklist and process the rounds in order. Round one marks AF, BE and CD. Round two marks AE, DF and BC. Give each entry a count rather than an ambiguous tick, so a second occurrence remains visible instead of disappearing beneath an existing mark.

Round three adds AD, CE and BF. Round four adds AC, BD and EF. Round five adds AB, CF and DE. Compare the resulting counts with the full checklist. Every one of the fifteen entries has a count of one. None has a count of zero, and none has a count greater than one.

This establishes both complete coverage and the absence of repeated pair identities. The first audit has already established that every participant appears once per round. Together, the two results verify this exact worksheet: five rounds, three unordered pairs per round, fifteen unique pairs overall, and one appearance per participant in every round. Neither audit needs to stand in for the other.

Read individual partner histories as a cross-check

A participant-by-participant reading provides another useful way to inspect the same arrangement. For C, the successive partners are D, B, E, A and F. Those are exactly the other five labels, with no repetition. For F, the successive partners are A, D, B, E and C. This reading can help locate a copying mistake that is difficult to notice within a row.

The remaining histories also agree with the target. B’s partners are E, C, F, D and A. D’s partners are C, F, A, B and E. E’s partners are B, A, C, F and D. Along with A’s history, these account for every participant’s five relationships. Each history contains all five other labels exactly once.

Treat these histories as a cross-check, not as permission to inspect only a convenient participant. A correct history for A says nothing conclusive about whether C and E have been paired twice elsewhere. The complete checklist remains the direct record of pair coverage. Individual histories are especially useful when you need to trace an identified problem back to its affected participants.

Introduce the deliberately faulty final round

Now create a separate faulty copy of the worksheet. Leave the first four rounds unchanged, but replace round five with AB CE DF. Label this copy clearly so that it cannot be confused with the verified version. The alteration retains AB while substituting CE and DF for the original final-round pairs CF and DE.

Run the first audit on the altered final row. AB uses A and B, CE uses C and E, and DF uses D and F. Every label still appears exactly once. The row therefore passes the participant-appearance test. This is not a failure of that test: the row genuinely satisfies the specific requirement that the first audit examines.

The faulty worksheet also retains five rounds and fifteen written pair positions. Its layout and totals therefore look unchanged. Those observations explain why a quick visual inspection can miss the defect. To determine whether all required relationships remain present, you must continue to the second audit and compare pair identities across the entire worksheet.

Identify the repeated and missing relationships

Process the faulty version against a fresh checklist. The first four rounds produce the same marks as before. When the altered final row adds AB CE DF, AB receives its first mark. CE receives a second mark because it already appeared in round three. DF also receives a second mark because it already appeared in round two.

Two required entries remain unmarked: CF and DE. Their only appearances in the correct example were in the original fifth round, and the faulty revision removed both. The diagnostic result is therefore precise: CE and DF are repeated, while CF and DE are missing. Record all four identities rather than describing the worksheet vaguely as having an incorrect final row.

The faulty plan contains fifteen pair occurrences but only thirteen distinct pair identities. CE and DF each occupy two positions; the other eleven present identities occupy one position each. The two extra occurrences displace the two missing identities. This shows why a total of fifteen written pairs cannot, by itself, prove that all fifteen required relationships are represented.

Repair the final round and repeat both audits

Restore round five to AB CF DE. Keep AB, replace CE with CF, and replace DF with DE. This repairs the exact defect while leaving the first four rounds unchanged. The corrected final row again contains A, B, C, D, E and F exactly once, so it satisfies the participant-appearance requirement.

Repeat the complete pair audit on the repaired worksheet. CE now appears only in round three, and DF appears only in round two. CF and DE each appear once in round five. Every other pair retains its single occurrence. The fifteen-entry checklist once again contains exactly one mark per entry, confirming that the repaired version restores complete, unrepeated coverage.

For this exercise, identify the repair as a return to AB CF DE, rather than treating an unexamined alternative as interchangeable. Any proposed rearrangement would need its own checks. Removing a repeated pair can accidentally give one participant two appearances in a round or introduce another duplicate elsewhere. A change is complete only when the resulting worksheet passes both audits.

Retain evidence for the exact version checked

Keep the verified five rows beside their completed checklist and note which version they describe. If you retain the faulty example, store it with its own findings: repeated CE and DF, missing CF and DE, and repaired final row AB CF DE. Audit marks belong to the specific rows examined; they should not be carried across an unchecked revision.

Loose cards can be useful as movable A through F markers if you assign each card a label on your worksheet. The Bicycle Disney six-deck bundle and Copag Elite playing cards are examples of physical cards listed by Republeq. The particular rank, suit, theme or material has no role in the pair-count proof. Plain slips of paper serve the same purpose.

Turning a paper pairing worksheet into an actual get-together raises other questions. The casual chess afternoon guide considers conversation and quiet play, while the coffee-table layout guide considers physical furniture clearance. Those are separate hosting and layout decisions; neither supplies or validates the fifteen-pair checklist here.

The exercise needs only written labels and a way to record counts. Product package quantities, card illustrations and material descriptions do not establish mathematical pair coverage, and no purchase is needed to complete this audit. Likewise, a correct pairing worksheet establishes nothing about availability, willingness, timing, scoring or physical arrangements. Those matters remain outside this fictional paper tutorial.

The final checked plan is AF BE CD; AE DF BC; AD CE BF; AC BD EF; AB CF DE. Its first audit confirms one appearance per participant in every round. Its second audit confirms exactly fifteen unique unordered pairs, each appearing once. Keeping those two questions separate makes the deliberately faulty row understandable and the repair demonstrable.