Did you solve it? Ace of spades | Mathematics

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Earlier today I set you the following two puzzles:

1. Deck dilemma

Your friend choses at random a card from a standard deck of 52 cards, and keeps this card concealed. You have to guess which of the 52 cards it is.

Before your guess, you can ask your friend one of the following three questions:

  • is the card red?

  • is the card a face card? (Jack, Queen or King)

  • is the card the ace of spades?

Your friend will answer truthfully. What question would you ask that gives you the best chance of guessing the correct card?

Solution It doesn’t matter. In all three cases, your chance of guessing the correct card is 1 in 26.

It’s a wonderful little puzzle because the result seems so counter-intuitive. Any question about the type of card gives you exactly the same help, which is to double your chances of getting the correct card.

Case 1. Once your friend replies, you will know if the card is red or black. There are 26 red, and 26 black cards, so you have a 1 in 26 chance of guessing the correct one.

Case 2. There is a 12/52 chance the card is a face card, and a 40/52 chance it isn’t. If your friend replies that it is a face card, you have a 1/12 chance of guessing the correct card, and if your friend replies it isn’t, you have a 1/40 chance.

Thus the probability of guessing the card when it is a face card is (12/52) x (1/12) = 1/52, and the probability of guessing the card when it isn’t is (40/52) x (1/40) = 1/52.

The overall probability of guessing the card is the sum of these two probabilities, which is 1/52 + 1/52 = 1/26

Case 3. The same argument applies. If the card is the ace of spades you will be told this fact by your friend, and this outcome has a 1/52 chance of happening. If the card isn’t the ace of spades, which has a 51/52 chance of happening, you must then chose 1 card from the remaining 51. This outcome gives you a probability of (51/52) x (1/51) = 1/52. Again, the sum of both possible outcomes is 1/52 + 1/52 = 1/26.

2. Heart is in pieces

The image below is a spade. Can you cut it into three pieces such that it is possible to reassemble the pieces and make a heart?

3.8 spade to heart Photograph: alex bellos

To be clear, what you are being asked to do is this: imagine the spade is made of card. Make two cuts to the card, thus cutting it into three pieces, and then reassemble the pieces without overlapping so that the pieces together make the shape of a heart, that is, the symbol of the suit of hearts. The cuts may, or may not, be straight lines.

Solution Cut as below, and then turn it around.

solution

I hope you enjoyed today’s puzzles. I’ll be back in two weeks.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

The first puzzle is adapted from Basic Probability, What Every Math Student Should Know by the eminent Dutch mathematician Henk Tijms. If you are interested in probability, and want an accessible, historical take, you might enjoy the following book by Henk’s son Steven Tijms: Chance, Logic and Intuition: An Introduction to the Counter-Intuitive Logic of Chance.

The origin of the second puzzle is either Sam Loyd or Henry Dudeney, who both published the puzzle around 100 years ago.

I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book. I also give school talks about maths and puzzles (restrictions allowing). If your school is interested please get in touch.

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