The Flipped Cards

Logic Level 2

Your friend presents you with a completely normal deck of cards. He then flips 13 of them over and then randomly places the flipped cards into different places of the deck . He then blindfolds you and bets you $10 that you can't get two decks of any size (that make up the whole deck) that have the EXACT same number of flipped-over cards. Should you take the bet?

Things to note:

1.Both sides of the card feel the same.

2.You haven't memorised where the cards are.

Finally, you can do anything with the cards as long as you don't take off the blindfold.

No Yes

4 solutions

Tin Le
Apr 21, 2019

How to win the bet:

Since there are 13 flipped cards, take any 13 cards carefully (avoid flipping any cards accidentally).
After taking 13 cards, flip it. Done! The 2 decks have the same amount of flipped-over cards.

Proof: Let's say $x$ is the number of flipped-over cards in the 13 cards which you choose.

Hence there must be $13-x$ flipped-over cards in another deck.

Since a card has only 2 sides, when we flip the 13 cards over, $13-x$ is the number of flipped-over cards.

Obviously, $13-x=13-x$ . Therefore, you win the bet no matter what value of $x$ is.

It’s a bit of a stupid trick question: any engineer would assume that the orientation of the cards with regard to “flipped” is normalized no matter how you orient the deck in 3-dimensions; just because you rotate one deck 180 degrees relative to the tangential plane of the planet doesn’t mean the definition of the original 13 flipped cards gets “flipped”.

David McCormick - 2 years ago

Jonathan Tse
Apr 14, 2019

Take off any 13 cards from the deck and flip the cards that you've just taken off the deck. This method works because let's say that you have 8 cards that are flipped over from the cards that you've just taken off the deck. Then in that case you would only have 5 in the other deck. If you flip over the deck of 13 cards, then you would have 5 flipped over cards, the exact same as the larger deck. This method works with any number of flipped over cards.

This is a stronger result than I was expecting. I didn't think that the player would be allowed to flip the cards himself.

Richard Desper - 2 years, 1 month ago

Richard Desper
Apr 18, 2019

Feels like the conditions need to be explained a bit. For example, when you say "two decks" are you saying two decks that combine to make the original 52-card deck? If I take two empty decks, that provides a trivial solution. My solution would be to create two decks of one card apiece. If you were to do that, you would have better than 50-50 odds of having the same number of flipped cards in the two decks. (This probability would be close to (3/4)^2 + (1/4)^2.) So if you have better than 50-50 odds of winning the bet, you should take it, right?

My reasoning was exactly like yours. The odds are in your favor if you just pick two decks of one card (this is what I understand by "two decks of ANY size"). For the other solution, some hint should be given that you are allowed to flip the decks and that the two decks combined make the original deck.

Gabriel Chacón - 2 years, 1 month ago

Okay I'll change that.

Jonathan Tse - 2 years, 1 month ago

I guess you have worded the question alright. With the simplest given solution i.e. taking 13 (equal to the number of flipped cards) cards out of the first deck just flip all of them over then place in the other deck will work regardless of the number of cards in the second deck (or even the first deck) even if they are a million, for argument sake.

Zahid Hussain - 1 year, 11 months ago

João Areias
Apr 23, 2019

I remember seeing this on Ted-ed https://www.youtube.com/watch?v=pnSw8g3DPHw

The Flipped Cards

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