Cards by email

Five cards

– Can we show you a new card trick, Abena?
– Go ahead!
– Pick five cards in this standard deck of cards.
– Any five?
– Let them be random. Shuffle the deck and pick five cards with eyes closed.
– OK. Then what?
– Email the five cards to Enu.
– How do I do that?
– Write down the value and suit of each card and email her.
– OK.
– Enu will email me four of the five cards, and me, brilliant as I am, will be able to deduce the fifth card!
– You are bluffing!
– Try me!

Problem creator: William Fitch Cheney.

5 Responses to “Cards by email”

  1. Michael says:

    This probably isn’t the intent of the puzzle but here’s my stab at it:

    Line #3 — “in THIS standard deck of cards” implies that the author is holding the deck of 52 cards.
    The recipient pulls five cards from the deck and writes them down. At no point does it mention that those cards are placed back in the deck. I think it’s safe to assume the author is, in fact, still in possession of the remaining 47 cards and thus knows which five cards are missing. When Enu e-mails the author four of the five cards, it is immediately obvious which card was the fifth.

    Again, probably not the intent of the puzzle but at least I *feel* like I’m following the rules.

  2. Michael says:

    OK here’s a method that requires planning but I think is more in line with the intent.

    For any given five cards at least two will be of the same suit. The first card shown of the four sent to me will be of the same suit as the one being held back. This will simply be agreed upon ahead of time. Furthermore, the one held back is the higher of the two suited cards.
    Example: 2 spade, J club, 10 heart, 4 diamond, Q spade
    Q spade is held back and 2 spade is the first card listed via e-mail. Receiving the 2 spade as the first card in e-mail tells me two things: (1) the card held back is a spade; and (2) the card held back is higher in value than a 2. (Values for this exercise are assigned as A=1, 2-10, J=11, Q=12, K=13.)
    How much higher is it?

    Well the remaining cards (3 of them) can be placed in order by value. Duplicate value cards are separated by suit such that spade > heart > diamond > club. By defining a hierarchy of suits the remaining three cards may easily be placed in a predefined order. Now, rearrange these cards as follows:

    Held back card is x cards higher than first card shown.
    Suits are listed BEFORE face values in e-mail and order of cards in e-mail is:
    123 — x=1
    132 — x=2
    213 — x=3
    231 — x=4
    312 — x=5
    321 — x=6
    Suits are listed AFTER face values in e-mail and order of cards in e-mail is:
    123 — x=7
    132 — x=8
    213 — x=9
    231 — x=10
    312 — x=11
    321 — x=12

    So again, from the above example, the hand is 2 spade, J club, 10 heart, 4 diamond, Q spade. The 2 and Q of spade are suited so the higher of the two is held back and the 2 is listed first in the e-mail. This indicates a held card of (2 spade + x). The Q’s value is 12 so we need to show +10. We do that by placing the remaining cards in the order of 2, 3, 1. Since the remaining cards are J club, 10 heart, and 4 diamond, we show 231 with (10 heart, J club, 4 diamond). Thus the e-mail is as follows:

    2 spade
    10 heart
    J club
    4 diamond

    Similarly, if the four cards listed via e-mail are (in order)
    heart 4
    club 6
    diamond 6
    heart club
    we notice that the order of the 6’s and Q are 123 (diamond being higher rank than club to break the tie). We also notice the suits are listed before the values. We know the hidden card is one higher than the 4 of hearts and is thus the 5 of hearts.

    Clear as mud?

  3. Michael says:

    Correction: Just above the last paragraph in my prior post the lines as follows:
    heart 4
    club 6
    diamond 6
    heart club

    …should read as follows:
    heart 4
    club 6
    diamond 6
    club Q

    Sorry. ‘Tis late.

  4. Alternatively: consider the denominations to be cyclic, with ace both low (below two) and high (above king). As Michael said, there will be at least one suit containing at least two cards A, B. What is more, it is possible to assign roles A and B suitably to two cards of the same suit so that B is no more than 6 ranks higher than A. E.g. with Michael’s example of “2 spade, J club, 10 heart, 4 diamond, Q spade” 2 is 3 ranks higher than Q (Q->K->A->2). So Enu holds the 2 back, lists the Q of spades first, and then orders the other 3 cards to signal “3”.

  5. Michael says:

    Richard’s point is a good one — and this allows a signal with even an image of the card (no trickery required by using the order of value and suit).

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