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Probability & planning · The puzzle notebook

Memory Match: calculate the turns left in a six-card finish.

Follow twelve possible layouts to the last pair, comparing two first-card choices with an explicit rule for every later turn.

H5 Free Fun · Published October 20, 2026 · 8 min read

A good chance of making your next pair is useful, but it does not tell you how many turns the rest of a Memory Match board will take. A mismatch can locate a partner, and a successful pair can leave another uncertain choice. To compare complete finishes, follow the information through every remaining turn.

Here is a small paper experiment you can reproduce. It compares two first-card choices, then uses the same finishing rule for both. The twelve-layout ledger gives an average of 47/12 remaining turns for one choice and 42/12 for the other. Those fractions belong to the specified experiment, not to every game.

Fix the information before choosing a card

Five pairs have already been collected from a sixteen-card board. The three remaining pairs are circle, triangle, and star. You accurately remember a circle at A1 and a triangle at A2. B1, B2, C1, and C2 have never been seen. All six unmatched cards begin face down; remembered does not mean currently face up.

    1 2 3 4
A   C T — —
B   ? ? — —
C   ? ? — —
D   — — — —

The letters stand for symbols: C is circle, T is triangle, and S is star. The four unknown positions contain C, T, S, S. Treat their twelve distinct arrangements as equally likely: four places for C, then three places for T; the two stars are identical. The actual cards never move during a finish.

Assume perfect recall of every revealed symbol, knowledge of the remaining pair counts, and correct deductions from them. A symbol forced by those counts is usable knowledge even before that card is turned up. This is a reasoning model, not an assumption about every player's memory.

Specify both starts and one shared continuation

A1-first: reveal the remembered circle at A1, then the unknown B1. Its partner was not already located. B1-first: reveal B1; if it is C or T, choose A1 or A2 respectively. If it is S, reveal B2. These choices use two different cards, just like a live turn.

After that first completed turn, both methods follow exactly this rule. Coordinates are ordered A1, A2, B1, B2, C1, C2; skip collected cards.

  1. Update the remaining information after each reveal. If every still-unseen card must have the same symbol, deduce those symbols. A deduction does not itself collect a pair.
  2. At a new turn, collect a completely known or deduced pair if one exists. Use the earliest coordinate with a known partner, and the earliest such partner.
  3. If no complete pair is known, reveal the earliest still-unknown coordinate. After seeing it, take a known or newly deduced partner if available; otherwise reveal the earliest other still-unknown coordinate.
  4. After a mismatch, choose Hide pair and continue before the next turn. Remember both symbols. Hiding is not an additional turn, and there is no third reveal in the failed turn.

The order convention resolves ties so that another reader can obtain the same ledger. We are comparing these complete procedures, not giving each method a convenient different follow-up when a result goes wrong.

Trace the layout T S C S to completion

Read the four letters in order B1, B2, C1, C2. Here B1 is T, B2 is S, C1 is C, and C2 is S; the remembered A1 and A2 remain C and T.

  1. A1 + B1: C and T mismatch. Hide them.
  2. A2 + B1: collect T.
  3. B2 + C1: S and C mismatch. Hide them. The only unseen card, C2, must now be S.
  4. A1 + C1: collect C.
  5. B2 + C2: collect S. This finish takes five turns.

Now restore the original knowledge, not the knowledge gained by reading that route. The B1-first procedure on the same hidden layout is:

  1. B1 + A2: collect T immediately.
  2. B2 + C1: S and C mismatch. Hide them; deduce S at C2.
  3. A1 + C1: collect C.
  4. B2 + C2: collect S. This finish takes four turns.

Both procedures collect exactly three pairs. The extra turn in the A1-first route is its first mismatch, making two mismatches rather than one. This is a complete accounting of that layout, not a claim that every initial mismatch is avoidable on an unfamiliar board.

Check all twelve possible layouts

Each four-letter string below describes B1, B2, C1, C2. The numbers include every completed turn from the initial choice through collection of the last pair. They exclude the earlier work that collected the first five pairs.

  1. CSST: A1-first 3 turns; B1-first 3 turns.
  2. CSTS: A1-first 4 turns; B1-first 4 turns.
  3. CTSS: A1-first 3 turns; B1-first 3 turns.
  4. SCST: A1-first 4 turns; B1-first 4 turns.
  5. SCTS: A1-first 4 turns; B1-first 4 turns.
  6. SSCT: A1-first 4 turns; B1-first 3 turns.
  7. SSTC: A1-first 4 turns; B1-first 3 turns.
  8. STCS: A1-first 4 turns; B1-first 4 turns.
  9. STSC: A1-first 4 turns; B1-first 4 turns.
  10. TCSS: A1-first 4 turns; B1-first 3 turns.
  11. TSCS: A1-first 5 turns; B1-first 4 turns.
  12. TSSC: A1-first 4 turns; B1-first 3 turns.

The A1-first column has two three-turn finishes, nine four-turn finishes, and one five-turn finish. Its total is 2 × 3 + 9 × 4 + 1 × 5 = 47. The B1-first column has six three-turn and six four-turn finishes: 6 × 3 + 6 × 4 = 42.

Divide by twelve because the layouts were assumed equally likely. The means are 47/12, about 3.917 turns, and 42/12 = 7/2 = 3.5 turns. The average difference is 5/12 of a turn. Seven layouts tie; in the other five, B1-first saves one turn under the shared rule.

Keep the measurements separate

For the first turn alone, A1-first matches in three of the twelve layouts; B1-first matches in eight. Those are probabilities, 1/4 and 2/3. The finishing averages above count turns, including later mismatches and known-pair collections. A probability difference is not automatically an average number of turns saved.

No actual finish lasts three and a half turns. Individual finishes here last three, four, or five whole turns. The fraction summarizes a model over layouts. Twelve new random rounds also need not produce each layout once or exactly reproduce either average.

Two paper checks

In layout CSTS, does starting at B1 finish in three turns just because the first pair matches?

No. Both starts take four turns here. After collecting C, B2 reveals S and C1 reveals T: a mismatch. The T pair and the inferred S pair still need their own collection turns. A successful first turn does not eliminate that later mismatch.

You deduce the final two unseen cards are stars. Are those cards already collected?

No. You now know their symbols and can select them as a certain pair, but both cards still have to be turned up in one completed turn. Deduction can remove uncertainty; it does not replace the collection action.

Use six paper cards or coordinate notes to work through another ledger row, always resetting to the stated initial knowledge before changing methods. These are partial-board paper positions, not diagrams that can be loaded into the live game. A retry of a real board preserves its layout but cannot erase what you have learned.

All twelve layouts and both complete procedures were checked independently, including deductions and turn counts. The comparison does not establish a universal strategy, an average for the full sixteen-card game, or a health or ability benefit. Different prior information, unreliable recall, or a different continuation rule requires a new calculation.