# Who is the liar?

Three friends are playing a game of memory, there are a total of 20 cards with 10 card pairs. They try to make the most pairs in a row, after everyone had a turn they make the following statements:

Person a: "I've made 10 pairs in a row, I'm perfect".

Person b: "I've made 9 pairs in a row, I'm not the best but I'm good enough".

Person c: "I've made 5 pairs in a row, I could do worse".

One person is lying, who is it?

• This problem isn't very clear what they are doing? Is there a procedure for playing this memory game? Commented May 27, 2015 at 9:55
• This problem isn't very clear what they are doing? Is there a procedure for playing this memory game? Commented May 27, 2015 at 9:59
• They are just playing memory (never played it I guess?), they are trying to make the most pairs in a row individually, after person a tried to make the most pairs in a row, it's person b's turn. Is it really that difficult? Commented May 27, 2015 at 10:01
• So A pulls cards off the deck until they get a miss and then they reset the deck for B? Commented May 27, 2015 at 10:04
• en.wikipedia.org/wiki/Concentration_%28game%29 look there for more information about the game: memory Commented May 27, 2015 at 10:06

## 3 Answers

well,

9 pair in a row is not possible .. because the two last (or first) cards have to be a pair as well (only a pair left), thus making it 10 pairs in a row .. so, B is lying.

• This might be me misunderstanding the question, but if he picked the last pair as first and last card he would get 9 pairs in a row. Commented May 27, 2015 at 9:28
• well, I could have misunderstood the question, as well, but .. I tought that players where removing two cards from the twenty, 10 times, trying to match pairs. If the card are removed, obviously, 9 in a row is the only impossible configuration.
– Den
Commented May 27, 2015 at 9:33
• It's a perfectly fine assumption, I believe :) Commented May 27, 2015 at 9:39
• You were right @Den It's just like a game of memory, got a pair, continue, no pair game over. Commented May 28, 2015 at 7:33

The answer is

Person B. If he got 9 pairs, there's only two matched cards left, meaning he's actually gotten all 10 pairs.

Either B or C, but it's impossible to know, if we assume that "I could do worse" means "someone did worse than me". B if "I could do worse" means "I didn't get the worst theoretical possible"

1. Let's assume A is lying and B and C are telling the truth. B states that he is not the best, which would mean he's lying, so without going further we can tell that A is telling the truth.

2. Let's assume B is lying. A's assertion doesn't conflict with B's, and if B did indeed make less that 5 pairs in a row, that would mean C is telling the truth.

3. Let's assume C is lying. With the above assumption he could be lying, and indeed be the one who did worst.