## Three Cards Trick

A forum for good logic/math puzzles.

Moderators: jestingrabbit, Moderators General, Prelates

kamenf
Posts: 9
Joined: Thu Apr 09, 2015 8:19 pm UTC

### Three Cards Trick

Do you know The Fitch Cheney Five Cards Trick? The following is a much harder twist of it.

Standard 52 cards deck. Magician is outside the room. Assistant asks random guy from the public to shuffle the deck and to select 3 (three) cards from it. Then asks another random guy to select 1 card from these three. Then Assistant lays out in a raw the three cards on the table - the last selected face dawn, the other two face up. Then Assistant calls Magician to enter the room. Magician looks for a while the three cards on the table and names the face down card.

What code is used by Assistant and Magician?

Take the following in mind:
1. There is no communication of any sort between Assistant and Magician.
2. Only the three cards arranged on the table give information to Magician.
3. Repeating the trick several times is safe.
4. It is possible this trick to be performed over phone, i.e. Assistant to tell arrangement of the three cards (naming only the two face up cards of course) to Magician.

dimochka
Posts: 77
Joined: Tue Apr 05, 2011 6:25 pm UTC
Location: A few different places->NYC->LA->NYC. He/Him/His please.

### Re: Three Cards Trick

Would it be correct to assume that the assistant will decide the order of the cards on the table (ie the face-down card could be in the middle, on the right, or on the left)? And that this would be communicated as such to the Magician when the trick is done over the phone?

kamenf
Posts: 9
Joined: Thu Apr 09, 2015 8:19 pm UTC

### Re: Three Cards Trick

dimochka wrote:Would it be correct to assume that the assistant will decide the order of the cards on the table (ie the face-down card could be in the middle, on the right, or on the left)? And that this would be communicated as such to the Magician when the trick is done over the phone?

Yes for both.

Posts: 455
Joined: Fri Nov 28, 2014 11:30 pm UTC

### Re: Three Cards Trick

To demonstrate this is possible, you should provide a list of things the assistant CAN do. If this trick is guaranteed to work, there must be a way to produce at least 50 distinguishable configurations with 3 distinct cards.

Clearly if order is the only thing you can mess with, you would not be able to do so since 3! < 50.
What other things can the assistant do? Do the cards have to be placed in a line? Can cards be rotated?
This is a block of text that can be added to posts you make. There is a 300 character limit.

kamenf
Posts: 9
Joined: Thu Apr 09, 2015 8:19 pm UTC

### Re: Three Cards Trick

Cradarc wrote:To demonstrate this is possible, you should provide a list of things the assistant CAN do. If this trick is guaranteed to work, there must be a way to produce at least 50 distinguishable configurations with 3 distinct cards.

Clearly if order is the only thing you can mess with, you would not be able to do so since 3! < 50.

This is the beauty of this mind blowing puzzle - it seams totally impossible
And all the required information is already given.
Spoiler:
Just take all in account.

Cradarc wrote:What other things can the assistant do?
Do the cards have to be placed in a line?

Spoiler:
Placing cards in a row (probably "in a line" means the same?) is sufficient. Though someone may find other solution if in repeating the trick it does not reveal easy observable differences.

Spoiler:
My solution is without rotation. Though someone may find a way to use rotation if it does not reveal easy observable differences in repeating the trick.

Cauchy
Posts: 602
Joined: Wed Mar 28, 2007 1:43 pm UTC

### Re: Three Cards Trick

Can you give an example of the communication the Assistant gives to the Magician over the phone in your phone example? I feel like this is the crux of the matter, maybe.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
Thanks, skeptical scientist, for knowing symbols and giving them to me.

kamenf
Posts: 9
Joined: Thu Apr 09, 2015 8:19 pm UTC

### Re: Three Cards Trick

Cauchy wrote:Can you give an example of the communication the Assistant gives to the Magician over the phone in your phone example? I feel like this is the crux of the matter, maybe.

Spoiler:
I think giving an example of such may mislead you. The reason is that although the principle of the encoding is somehow similar the medium is different - one is the space while the other is time. The only thing I can say is that over phone for each card assistant use the same words in all executions of the trick, eq. " Last card is face down, middle card is Ace of spades,...". I mean no manipulations with wording itself is used and actually computer synthesized voice could be used for no doubt. I put this point only to note that principles and mathematics used to encode bits of information needed to calculate the card can be used for over phone presentation of this trick.
Hmm... by reading what I've wrote it seams a big spoiler to me

Cauchy
Posts: 602
Joined: Wed Mar 28, 2007 1:43 pm UTC

### Re: Three Cards Trick

It sounds like your solution doesn't even really rely on there being three cards. The assistant could just as easily put only the face-down card on the table, or report to the magician "I put a card face down on the table".
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
Thanks, skeptical scientist, for knowing symbols and giving them to me.

kamenf
Posts: 9
Joined: Thu Apr 09, 2015 8:19 pm UTC

### Re: Three Cards Trick

Cauchy wrote:It sounds like your solution doesn't even really rely on there being three cards. The assistant could just as easily put only the face-down card on the table, or report to the magician "I put a card face down on the table".

If someone finds a stable solution of the problem with less then three cards I would be really surprised. At the end having a solution with one card like placing it on one of 52 imaginable positions on the table or rotating it to multiples of about 3.5 degrees to indicate card index are first not stable and second not serious.

Cauchy
Posts: 602
Joined: Wed Mar 28, 2007 1:43 pm UTC

### Re: Three Cards Trick

Where do you draw the line, then? Since the cards given are random, the Assistant has two arbitrary cards to work with and must be able to encode any of the other 50 cards somehow. As Cradarc said, there are 3! = 6 orderings of the cards, so somehow more information has to be encoded than in just the order. This is where things start to seem fishy. Where are you encoding the other information? I'm having a hard time believing that wherever you store this other information will sound significantly more serious than "the Assistant rotates the face-down card by a/100 degrees clockwise, where a from 1 to 52 corresponds to the 52 cards of the deck in an agreed upon manner", or "the Assistant introduces a pause between her third and fourth words of length (a+50)/100 seconds, where a again indicates the face-down card" for the phone version.

Even if the Assistant got to choose which of the three cards was face down, it still wouldn't be possible based on just the ordering, since there are 52 choose 3 hands the Assistant could be dealt but only 6*(52 choose 2) set-ups they can show the Magician, and 52 choose 3 > 6*(52 choose 2).
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
Thanks, skeptical scientist, for knowing symbols and giving them to me.

phlip
Restorer of Worlds
Posts: 7572
Joined: Sat Sep 23, 2006 3:56 am UTC
Location: Australia
Contact:

### Re: Three Cards Trick

Indeed. I can come up with some other sources of information... like, in a standard pack of playing cards, the odd-valued number cards (including aces) of each suit other than diamonds are orientable (by which I mean non-rotationally-symmetric)... but you can easily just use a deck where all the cards are orientable, both front and back. Then you can get another 3 bits just from each card being possibly upside-down. Make the orientability subtle (like it already is on the cards that are orientable in a standard deck) and no-one would notice. That gets you all the way up to 48 possibilities, which is almost enough already. Or you could lay out the 3 cards slightly overlapping each other... and then for each of the two overlaps, either the left card is on top or the right card is on top, for another 2 bits. Or exactly where you put the remaining 49 cards in the deck, presumably in a pile somewhere on the table, could matter. There's plenty of options.

But it's difficult to know exactly what is "in scope" for this as a logic puzzle as opposed to a real-world magic trick. Hard to tell what you'd accept as a real solution, and what you'd consider "cheating"...

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]

Moonbeam
Posts: 292
Joined: Sat Dec 08, 2007 5:28 pm UTC
Location: UK

### Re: Three Cards Trick

There must be a way of providing more info when the assistant reads the cards out to the magician.

I'm assuming that the following scenario WON'T work:

Spoiler:
The assistant lays out the order of the cards.
The assistant then uses another independent "helper". He writes down exactly what the helper should say to the magician.
The helper then reads what the assistant wrote down, but does it in whatever manner he so chooses, taking as long as he wants in between naming the cards with whatever pitch of voice he wants, etc.

Would the assistant still be able to pull the trick off - I'm assuming no ????

PeteP
What the peck?
Posts: 1451
Joined: Tue Aug 23, 2011 4:51 pm UTC

### Re: Three Cards Trick

Easy for the phone
Spoiler:
The assistant just needs to say 2 sentences, the pause between the sentences is exactly 1 second+x milliseconds long where x is a number identifying the card. The trick is that both are highly advanced cyborgs who can do that.

Or alternatively for placing them on the table. 6 possible ways to order them, you just need 3 bits more (48 is enough since we know 2 of the 49 cards). 2 bits from leaving a gap or not leaving a gap (with 2 possible gaps). 1 bit from placing them in a row so that their short edges meet or so that their long edges meet.

jaap
Posts: 2094
Joined: Fri Jul 06, 2007 7:06 am UTC
Contact:

### Re: Three Cards Trick

I have read of a legitimate variant of this trick, but that involved 5 cards:
- 5 cards are freely chosen from a deck of 52.
- The Magician's assistant chooses one of these 5 cards, and lays the other 4 face up in a row.
- Those 4 cards are read out by a spectator to the magician on the phone.
- Magician determines the chosen card.
In this case you have the 4!=24 ways of ordering the cards, but there is also the fact that it is the assistant who chooses which of the 5 cards to use which restricts things enough for those 24 options to be sufficient to determine the card.

I have no clue about the version posted in the OP. It might involve the direction of the row (using overlapping cards) for the suit, and the location on the table for the value. That's not much of a logic puzzle though.

qetzal
Posts: 862
Joined: Thu May 01, 2008 12:54 pm UTC

### Re: Three Cards Trick

i think kamenf gave the clue above. Besides the order of the cards on the table,
Spoiler:
there's also the order that the Assistant names them. That allows 36 different combos. And that should be plenty because there are really only 17 permutations needed: 13 for the rank (2-A) and 4 for the suit.

phlip
Restorer of Worlds
Posts: 7572
Joined: Sat Sep 23, 2006 3:56 am UTC
Location: Australia
Contact:

### Re: Three Cards Trick

qetzal wrote:
Spoiler:
And that should be plenty because there are really only 17 permutations needed: 13 for the rank (2-A) and 4 for the suit.

Spoiler:
13 for the rank and 4 for the suit, but that makes 13*4 total permutations needed, not 13+4. The magician needs to be able to learn both, not just one. Which means you need at least 52. (Well... 50, since two cards are known.)

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]