# Rabbit’s Card Puzzle

31/08/2010 13 Comments

Two people sit down at a table, each with a shuffled pack of standard playing cards in front of them.

They each turn over the top card and place it face up on the table. If they have the same card they shout “SNAP!”

For this to occur both number and suit must match. Four of Spades does NOT match with Four of Hearts.

The chances of this happening on the first go is 1 in 52.

Now more people want to join this fantastic game. Each time a new player joins they bring their own pack of shuffled cards.

i.e. When there are 4 players, 4 cards are turned over. If there is a pair anywhere among the 4 cards they all shout “SNAP!”

As the size of the group increases, the chances of shouting “SNAP!” increases.

At some point the group becomes so large that it is more likely than not that “SNAP!” will be shouted when they each turn their first card over.

What is the smallest number of players required to make it more likely than not that “SNAP!” will be shouted when they each turn their first card over?

I will donate £10 to the chosen charity of the person with the first correct answer.

They need a rough reason though. You can’t just guess 1,2,3,4 etc. until you get the right number.

Please put your answer in the comments on this post, not on Twitter.

Good luck!

RedEaredRabbit

25 decks

Sorry, it’s not 25.

I didn’t say 25! I said 27. Do I win?!?

27?

27 is my best (and only guess). My rubbish reasoning is that once you’ve got more than 26 players (ie half a pack’s worth) you’re more likely to get a snap than not.

SPOILERS BELOW – DO NOT READ IF YOU ARE YET TO WORK IT OUT

The answer is 9. It’s an alternative version of the matching birthdays problem, but with a denominator of 52. We need 1 – 51/52 x 50/52 x … to be greater than 0.5. This happens with 9 players. Now I’ve spoilt it for everyone else. You are Blaise Pascal and I claim my 10 bob note.

RIGHT!!!! Well done!!

I think the answer is eight.

With 2 players, the chance is 1/52. With 3, it’s 1/52+(51/52*2/52). With 4, it’s 1/52+(51/52*2/52)+(50/52*3/52). And so on. When you get to eight, you crunch the numbers and get 0.516 or so (assuming I haven’t been hitting the wrong buttons on my calculator), which is just over 50%, or the required one in two chance of people shouting SNAP!

(If I’m right, then £10 to the Refugee Council, please.)

Very close, but not right.

16? By that point the probability is 1 in 7.5, but there are 16 cards being turned over so it is morelikely that not a SNAP will occur?

27

Weird – the other comments weren’t showing till I posted! I take it back – there are more reasoned arguments.

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