## Math Card Tricks: I Know the 5th Card

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Ξ and I were recently shown a card trick by a colleague and a former student. It was a two-person show, with the student playing the role of “Assistant” and our colleague playing the “Mind Reader”. This is what we saw.

The Mind Reader left the room, and the Assistant explained that she could communicate with the Mind Reader telepathically. We chose five cards (total) from a standard deck of 52, then showed them to the Assistant. Here are our cards:

She picked one of the five (the 2), told us to remember it, hid it, then arranged the remaining four cards in a row, like so:

The Assistant then told the Mind Reader to enter (why didn’t she just “think” him back in?). The Mind Reader studied the cards and proudly proclaimed that the last card was the 2. Ta-da!

After watching the trick two more times, it wasn’t hard to figure out how the suit of the hidden card was signaled—the Pigeonhole Principle guarantees at least two cards in the same suit, so choose one of the cards from the repeated suit and place it first. But how to indicate the value? (Go ahead and think about it for a little while.)

The first step is to order the deck, typically from A-K within a given suit, and then order the suits (the contract bridge ordering of clubs, diamonds, hearts, spades being common). Now given any, oh, say three cards, we can label them HIGH, MIDDLE, and LOW. Why? Because there are now 3! = 6 permutations of these cards, and 6 is the magic number.

Next, suppose we’re given five cards, two of which are of the same suit. Using the above example, we have the A and 2 of hearts. Hide the higher of the two with one caveat: if the cards are more than 6 apart, use the lower card. (Here’s why: If we add modulo 13, the lower card is actually no more than 6 “above” the higher card.)

Finally, arrange the other four cards so that the first card matches the suit of the hidden card, and the last three cards indicate how far the hidden card is “above” the first, using the following code:

LMH: +1
LHM: +2
MLH: +3
MHL: +4
HLM: +5
HML: +6

Of course, we can always vary the location of the “suit indicator”, especially if we’re going to show the trick several times, and we could even vary the ordering of the cards, or the meanings of LMH, etc. A nice feature of this trick is that it is incredibly difficult to determine the “signal” that the Assistant uses to communicate the card to the Mind Reader, and if we only do the trick once, chances are nobody will figure it out.

According to Colm Mulcahy, this trick is known as “Fitch Cheney’s Five Card Trick”, and first appeared around 1950. (Read the Math Horizons article here. I first saw it at the h2g2.) In the article, Mulcahy points out that this trick can be generalized to a 124-card deck. He also gives a variant that uses up/down (right-side up and upside-down) to encode the addition. After you’ve read the article, check out his other tricks (he does a column every month).

Now go forth and amaze your friends!

### 2 Responses to “Math Card Tricks: I Know the 5th Card”

1. jd2718 Says:

I know this one. And so does a professor from a college 3 hours north of here. We run into each other at math teacher conferences. Usually our meetings are signaled by a yell: “John!” I look up. “What card does ______ have?” I walk over to see what four cards Jim is holding…

We’ve been doing this for years, never fails to amaze. But here’s the kicker. We eventually swap roles, and I talk to the people watching “just give me four cards” (and I need to pay attention, pigeonhole is no longer in play) and stick a joker in your pocket!

4h, 5c, Jd, 6h

Jonathan

2. Phil Galati Says:

Great Trick!