h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: The Five Card Trick - A2496873
Author: Gnomon: Pointing out the way, The shadow on the sundial; Samhain awaits us [3NY] - U151503

A mathematical card trick.

Nice work

"5. Remember to let the Great Marvo take all the credit, while grinning inanely."



One quick (very, very minor) question:

Is there any specific reason why the order of suits ("Some boring but important details") is Hearts Clubs Diamonds Spades? As a (very bad) Bridge player, it would make sense to me for the order to be Spades Hearts Diamonds Clubs. If the order is simply at the discretion of Mitzy and Marvo, perhaps that could be stated?

Editorial things:

FOr example -> For example,

Mitzy is sometimes called Mitzi (surely she's not so inane she can't spell her own name? )

The order of the suits is not important. If the one you stated is the norm in the card-playing world, it will be easier for people to remember so I may change it to that.

It's the rule for Bridge (Spades ranking highest, Clubs lowest). I can't speak for other card games where suit order may be important.

Thanks for spotting the typos.

I've changed the order to the Bridge order and added a note to say that any order will do.

I now have a vision of Mitzy in her sequined leotard and a feather in her hair sitting across a Bridge table from Marvo in his cape and top hat

Nice one, Gnomon

This was on my list of entries to write one day! I've got some interesting info on the maths behind it, but it's at home, so I'll post it another day. The point is that the trick is difficult to work out as people can't immediately see how the assistant can give enough coded information purely in the order of 4 cards. I think I've got some history on it too.

One thing that springs to mind is that the audience will soon guess that the first card the assistant plays is the same suit, so she should vary it by arrangement for each performance.

Now Marvo looks at the other three visible cards.

Icy

Nice one, Gnomon. Maybe I'll try that one day.

Typical, she does all the work, he gets all the credit.

You'll have the Magic Circle after you now, but great entry!

Like it.

I may be being dense here, but I take it the Great Marvo is not a real magician?

The order of suits in bridge is, lowest to highest, alphabetical order: CDHS. That may or may not be worth mentioning.

Gnomon,

I dug out my notes on this trick. Feel free to use any of this if it suits.

The trick was invented by US mathematician William Fitch Cheney Jr (1894-1974), and it was first published in 'Maths Miracles', by Wallace Lee (Seeman Printery, 1950)

It's more of interest to mathematicians than magicians, as it's a very accessible scenario for problems in the branch of mathematics known as information theory. In it's purest sense, the trick involves one person handing 4 cards to another person, one at a time. The 5th card can then be determined by the order in which they were received. The point of the trick is that on the face of it, there are only 4! or 24 combinations of those 4 cards, so it would only work with a pack of 24+4, or 28 cards. There is obviously some unseen information that is being passed, but this is not apparent.

This is why it's more of a theoretical trick than a practical one - there are all manner of ways in which a magician's assistant could send extra information. You mention Mitzy winking, coughing or tapping her nose, as each of these could pass another bit of information. I know you say she doesn't, but the point is that she could very easily, and there's no way an audience could tell. In short, a magician could probably work out a sequence of events which would identify a card without having to have any other cards passed at all.

Some more on the maths. There is some superflous information which is being passed, and in theory, the trick can be performed with a deck of as many as 124 cards, but the algorithm used will be different.

If anyone intends to go out and perform this, then I'd suggest they practice it at least 20 times before they try it in front of an audience. Getting it wrong just once can be a career-defining moment for a magician.

Hope this helps.

Icy

Considering Mitzy's doing all the hard work, I hope Marvo doesn't do something foolish like saw her in half!

Good entry, Gnomon. Well-organized, fun to read. My only suggestion: if there are previous entries in computer science or infomation theory about encoding data or how many bits it takes to encode how much information, it might be nice to link to them.

You've thrown me Icy - first you claim that the deck can only have 28 cards, then you say 124. I know why the first is wrong - Mitzy selects the hidden card to be of the same suit and 1-6 places above one of the others, so while she can only indicate 24 different cards by her arrangement of the others, the hidden card is one of those 24.

But where do you get 124 from and what would the algorithm be?

I'm away at the moment but will deal with these points when I get back.

Bagpuss, for the deck of 124, I'm inclined to refer you to Michael Kleber's article 'The Best Card Trick', published in Mathematical Intelligencer 24 #1 (Winter 2002), as it's a little complex to explain. It involves the Birkhoff-von Neumann theorem and Hall's Marriage theorem, if that helps.

As I understand it, the assistant can choose the sequence of the hidden card as well as the others, so that would appear to make it possible with 5!+4 or 124 cards, but it may be a little more complicated than that.

Icy

Ah. Sounds reasonable.

The article fails to mention the obvious, but then many 'magic' tricks work by the observing people seeing but ignoring the implications of obvious.

Obvious fact #1 In any set of 5 cards, there must always be at least two cards of the same suit.

OF #2 A card can never be more than six away from any other card.

I think it would be amusing to link to the Edited Entry on Alogrithms somewhere. The word alogrithm doesn't need to be in the article, if you don't want.

The unobvious fact is that Mitzi gets to choose which of the five cards she puts face down. If she wasn't allowed to do this, the trick would not be possible as stated.

I'll try and add a few details over the next day or two to take what you've all said into account.

Reminder to self to work on this.

I've added a section "A Mathematical Trick" at the end. I'd be grateful if someone with a mathematical brain could check it over.