Aug 13, 2022Kuina-chan


I will verify that there are situations in the card game "memory game" where it is better to make mistakes on purpose.

1Memory Game Winning Percentage

Memory game, a card game played by two players, is a major game called "neurasthenia" in Japan. It is a game that tests the memory of the players, with the rule that one of the players turns over two cards, and if they have the same number, they get it; if they have different numbers, they put them back face down and it's the other player's turn.
Many people may be playing with the tactic shown in Figure 1-1.
  1. If the same number exists among the cards that have been turned over once, you choose those two cards.
  2. If not, you choose a card that has never been turned over.
  3. If the same number as the card you chose is among the cards once turned over, you choose it.
  4. If not, you choose from among the cards that have not been turned over.
Figure 1-1: Tactic Of Memory Game
In fact, when both players with good memory use this tactic, there is a big difference in the winning percentage between the first and second player. Let's take a look at it below.

1.1For Two Cards



If both players have a good enough memory and never forget, the memory game can be regarded as just a game of probability. Assuming that the players use the tactics described earlier, let's find the winning percentage for both.
If a memory game is played with two cards, the first player chooses those two cards, so obviously the first player wins 100% of the time (Figure 1-2).
Winning Percentage For 2 Cards
Figure 1-2: Winning Percentage For 2 Cards
In this example, spade and heart cards are used.
There are two patterns: the first player draws the A of Spades the first time and the A of Hearts the second time, or the first player draws the A of Hearts the first time and the A of Spades the second time, but the first player wins in both cases.

1.2For Four Cards



If the players play a memory game with four cards, covering all possible patterns, the result is that the first player wins 1/3 of the time and the second player wins 2/3 of the time(Figure 1-3).
Winning Percentage For 4 Cards
Figure 1-3: Winning Percentage For 4 Cards

1.3Up To 12 Cards



If you use six cards, the number of possible patterns increases and the calculation is more complicated, but the winning rate for the first hand is 7/15 and for the second hand 8/15.
As a result of steady calculations since then, the winning percentage up to 12 cards was as shown in Table 1-1.
Table 1-1: Winning Percentage Up To 12 Cards
Number Of Cards Winning Percentage Of The First Player Winning Percentage Of The Second Player Draw
2 100.00% 0.00% 0.00%
4 33.33% 66.67% 0.00%
6 46.67% 53.33% 0.00%
8 60.00% 26.67% 13.33%
10 45.40% 54.60% 0.00%
12 36.82% 50.83% 12.35%
As you can see, there is a big difference in the winning percentage between the first and second player depending on the number of cards.
As the number of cards increases, it becomes more difficult to calculate the winning percentage, but the winning percentage of the two players approaches 50% each.

2Cases Where You Should Intentionally Make Mistakes In The Memory Game

Now here's a tactic to make a deliberate mistake(Figure 2-1).
  1. If the same number exists among the cards that have been turned over once, you choose those two cards.
  2. If not, you choose a card that has never been turned over.
  3. If the same number as the card you chose is among the cards once turned over, you choose it.
  4. If not, you choose a card that has already been turned over to intentionally make a mistake.
Figure 2-1: Tactic Of Making Mistakes On Purpose
The only thing different from the first tactic is the last part. If a player opens a card unnecessarily, it will give a clue to the other player, so the tactic is to intentionally make a mistake to pass the turn to the other player (i.e., pass).
Let's call the first tactic "tactic A" and this tactic of deliberately making a mistake "tactic B."
If there are no cards already turned over, the players cannot pass, but to simplify the story, the rule is that they can always pass, and the result of calculating the win percentage with 12 cards is shown in Table 2-1.
Table 2-1: Winning Percentage For 12 Cards
The Tactic Of The First Player The Tactic Of The Second Player Winning Percentage Of The First Player Winning Percentage Of The Second Player Draw
Tactic A Tactic A 36.82% 50.83% 12.35%
Tactic A Tactic B 54.38% 31.52% 14.10%
Tactic B Tactic A 50.09% 37.27% 12.64%
Tactic B Tactic B 19.35% 66.22% 14.43%
When both players took tactic A, the win percentage of the first player was only 36.82%, but by deliberately making mistakes, the first player was able to increase his win percentage to 50.09%.

3Conclusion

This experiment confirmed that, depending on the number of cards used in the memory game, a player's winning percentage may be improved by deliberately making mistakes.
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