Card Trick Cipher Solution

This is one of the possible solutions to the Alice and Bob card trick, devised by John Bohannon.

Alice gets 5 random cards. The convenient thing about 5 cards is that at least 2 of the cards will have the same suit (since there are only 4 suits). She chooses 2 cards of the same suit and designates one of them as the Mystery Card and one of them as the Base Card. First she must figure out the difference between the cards' identities; the Mystery Card must be no more than 6 mod 13 steps away from the Base Card.

For example: Let's say that she has a 3-Clubs and a 7-Clubs. That one is easy. Counting up, it takes 4 steps to get from the 3-Clubs to the 7-Clubs. So the 3-Clubs will be the Base Card and the 7-Clubs will be the Mystery Card.

Harder example: Let's say that she has a 3-Clubs and a 10-Clubs. It takes 7 steps to get from 3 to 10, which is more than 6. But starting from the 10-Clubs: 11-Clubs (also known as the Jack of Clubs), 12-Clubs, 13-Clubs, 1-Clubs (remember, this is mod-13, like counting on a clock), 2-Clubs, 3-Clubs. That took exactly 6 steps. Perfect. So in this case, the Base Card is the 10-Clubs and the Mystery Card is the 3-Clubs.

Once Alice has chosen the Mystery Card, she must now order the remaining 4 cards for Bob.

The first (leftmost) card will be the Base Card. The next 3 cards encode the number that Bob will add to the Base Card to figure out the Mystery Card.

To see how the encoding works, let's switch to Bob's perspective once Alice hands him the 4 ordered cards.

Bob figures out the Mystery Card identity like this:

  1. He knows that the suit of the Mystery Card is the same as the suit of the Base Card. He's halfway to the solution already!
  2. Now, figure out the relative rank of the 3 remaining cards (low, middle, high). Aces = 1, and alphabetical rank of suit is tie-breaker, i.e. the rank of (4-Clubs, 4-Hearts, 4-Spades) = (low, middle, high).
  3. Now think of the cards as 1,2,3 rather than low,middle,high. (In math-speak, convert the cards into a 3-digit base-3 number.) So for example (13-Spades, 3-Diamonds, 8-Clubs) = (high,low,middle) = (3,1,2) = 312.
  4. Now translate that 3-digit number to the number you need to add to the base card to find the mystery card. The mapping is simply their ascending rank in size:
    5. 1:  123
2:  132
3:  213
4:  231
5:  312
6:  321

An easier way to think of it: There are only 6 different numbers that you can make by permuting the digits 1, 2, and 3. The smallest is 123, so that has a rank of 1. The largest is 321, rank of 6. And so forth.

The order of the cards gives you one of these 6 numbers. Now just add that number to the number of the Base Card (mod-13, and don't forget that Ace=1), and you've solved the identity of the Mystery Card.

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