Math Card Coding Mindreading Trick 1 Accomplice leaves

Math Card Coding Mindreading Trick 1. Accomplice leaves room. 2. Audience picks out 5 arbitrary playing cards from deck. 3. Presenter arranges 4 (carefully chosen) face-up, and one face down. 4. Accomplice returns to the room, and announces the face-down card. This trick relies on three nice ideas from discrete mathematics: • pigeonhole principle 5 pigeons (cards) in 4 holes (suits) means at least two pigeons are in the same hole - i. e. , at least two of same suit. • mod arithmetic Arrange cards on 13 -position clock. One of the same-suit pair can be used as a “base card” occurring at most 6 cards before the other card, which is chosen to be the hidden card. • permutations There are 6 distinct arrangements of the 3 remaining cards - the one chosen indicates whether to add 1, 2, 3, 4, 5, or 6 to the base card. Base card gives suit of hidden card, and starting rank on “card clock” +1 Permutation is fifth possible (123, 132, 213, 231, 312, 321) +2 +3 +5 +4 +5 +6

Idea #1 pigeonhole principle + = corollaries: at least two of same • socks • others • among five cards, at least two have same suit

Using Idea # 1 Base card gives suit of hidden card. = Idea #2: Card Clock (mod arithmetic) +5 Base card also gives starting point on card clock. Now only need to signal “ADD 5”

Idea #3: permutations • There are exactly 6 permutations of 3 objects • There are 3 objects to be ordered (3 middle cards) • I can order the permutations systematically perm-1, perm-2, perm-3, perm-4, perm-5, perm-6 • Depending on which permutation, I can indicate to add 1, 2, 3, 4, 5, or 6, to the base card. 1 2 3 = +1 1 3 2 = +2 2 1 3 = +3 2 3 1 = +4 3 1 2 = +5 3 2 1 = +6 if first card is the lowest, then add 1 or 2, depending on whether the remaining two cards are in or out of order if first card is the mikddle, then add 3 or 4, depending on whether the remaining two cards are in or out of order if first card is the highest, then add 5 or 6, depending on whether the remaining two cards are in or out of order

Additional Considerations (a) What if two of the three middle cards are the same rank? Answer: Use a natural ordering on suits, with clubs < diamonds < hearts < spades, so that 7 D 7 H 7 C would indicate “add 4”, for example. (b) What if the base card is J and hidden card is 5? 1. You can’t reach 5 from J in 6 or fewer steps. But then this means you can reach J from 5 in 6 or fewer steps. Choose 5 as the base card, J as hidden. This is the job of the main presenter from the two cards of the same suit chosen, figure out which card should be the base, from which the hidden card can be reached in at most 6 steps. Notes The first page is a summary page that can be printed out for handy reference. The next four pages are useful for a presentation of the trick to an audience. This trick has been around in the recreational mathematics and teacher communities for a while, and I do not know the origin. This presentation prepared by Lenny Pitt, pitt@uiuc. edu