Memory Match: which card should you reveal first?
Compare two reveal orders on a six-card endgame, count the possible matches, and separate knowledge from luck.
You remember one circle and one triangle, but neither partner has been found. Should you start the next turn with a card you recognize, or reveal something unknown first? On the position below, revealing an unknown card first gives you a better chance of completing a pair during that turn.
The reason is the choice you retain for the second reveal. If the first card matches a symbol you already know, you can select its remembered partner. If it does not, you can explore another unknown card. We can check this advantage by counting the possible hidden layouts instead of judging it from one lucky round.
Set up one precise information state
Imagine five pairs have already been collected from the sixteen-card Memory Match board. Six unmatched cards remain. Rows are A–D from top to bottom; columns are 1–4 from left to right.
1 2 3 4 A ● ▲ — — B ? ? — — C ? ? — — D — — — —
The circle at A1 and triangle at A2 are remembered symbols, not necessarily face-up cards. All six unmatched cards are face down before the turn starts. A question mark means never revealed; a dash means already matched and unavailable. This is a paper knowledge map, not a board-loading feature.
The remaining pairs are circle, triangle, and star. Since one circle and one triangle are already located, the four unknown positions contain exactly this collection:
● ▲ ★ ★
Assume your remembered locations are correct and you have no further clues about those four positions. Treat every arrangement of the remaining cards among them as equally likely. The probabilities below describe this information state; the actual board stays fixed during the round.
Option one: reveal the remembered circle first
Choose A1. Its circle is already known, so this reveal gives no new symbol information. The second circle must be in one of B1, B2, C1, or C2. With no clue distinguishing those positions, choosing any one of them has a 1 in 4 chance of matching.
A2 cannot match: it is a known triangle. Choosing A1 again is not another chance, because you must reveal a different second card. Starting with the remembered triangle instead gives the same 1 in 4 chance, since its partner is also among the four unknown positions.
Option two: reveal B1 first, then use what it shows
B1 is unknown. Its four equally weighted possibilities are one circle, one triangle, and two stars. Choose the second card according to the first reveal:
- Circle, probability 1/4: choose A1. You have a certain match if your memory is correct.
- Triangle, probability 1/4: choose A2. Again, you know the matching location.
- Star, probability 1/2: neither remembered card can match. Choose another unknown position, such as B2.
In the star case, B1 has removed one star from the unknown collection. B2, C1, and C2 contain one circle, one triangle, and the other star. B2 therefore has a 1 in 3 chance of matching B1. The two remembered cards are excluded because their symbols are already known to differ.
Combine the branches to find the chance of a pair this turn:
1/4 + 1/4 + (1/2 × 1/3) = 1/2 + 1/6 = 2/3
That is 2 in 3, compared with 1 in 4 for starting with a remembered single. The two methods start from the same six-card state. Their difference comes from adapting the second reveal to new information, not from changing the shuffle or getting an extra reveal.
Check the count without decimals
There are twelve distinct symbol layouts for the four unknown positions: choose the circle’s location in four ways, then the triangle’s location in three ways. The two remaining positions contain identical star symbols.
For the remembered-circle method, three of those twelve layouts put its partner at your chosen unknown position: 3/12 = 1/4. For the B1-first method, three layouts show a circle at B1 and three show a triangle there. Those six all give an immediate known-partner match. Among the other six layouts, B1 is a star; two also have a star at B2. That makes eight matching layouts in total: 8/12 = 2/3.
This exact count is more informative than trying twelve random rounds and expecting exactly eight matches. A small random sample can vary. The fraction comes from all compatible layouts under the stated equal-likelihood assumption.
Turn a mismatch into a usable next move
Suppose B1 reveals a star and B2 reveals a circle. The turn is a mismatch, but the circle’s partner is already known at A1. You cannot reveal A1 as a third card this turn. Read the locations, choose Hide pair and continue, then select B2 and A1 on the next turn to collect the circle pair.
The mismatch leaves useful information: B1 is a star, A2 is a triangle, and the two still-unknown positions C1 and C2 contain one star and one triangle. Keep that information attached to the coordinates. Cards do not move when a mismatch is hidden.
Self-check: update the unknown pool
Return to the original knowledge map for each question.
B1 shows a star. Is the next unknown card’s match chance 1/5, 1/4, or 1/3?
It is 1/3. Five other unmatched cards remain, but A1 is known to be a circle and A2 a triangle. Only B2, C1, and C2 can hold the matching star, and one of those three does. This answer uses the remembered information; counting every face-down card would ignore it.
B1 shows a triangle. Should you guess at B2?
Choose A2 instead. Its triangle is already located, so it is a certain match under the accurate-memory assumption. The 1/3 calculation belonged to the star branch; it is not a fixed chance for every second reveal.
Know what the comparison establishes
We compared two specific methods for maximizing the chance of a match on the next turn in this particular state. We did not prove the best strategy for every board or the fewest expected turns for an entire game. If a complete pair is already remembered, you can collect it directly. If you are unsure of a remembered symbol, the certain-match branches no longer apply.
Use the calculation as a habit: distinguish known locations from unknown ones, reveal information, then update the possible partners before choosing again. The game is untimed, so you can inspect a mismatch at your own pace. One successful guess is not evidence of better memory, and this exercise makes no health or ability claims.