Thanks to Patrick Honner and the inimitable Card Mulcahy for pointing me to the origins of this trick. It’s the Fitch Cheney 5-Card Trick.
I was eight, maybe nine, at the kitchen table with my dad. He was describing a professor who claimed his grad student is psychic.
“The professor asks you to draw five cards from the deck,” my dad explained. “Then the professor shows four of them to his grad student, who guesses the fifth one.” My dad paused for emphasis. “He gets it right every time.”
“But he’s not really psychic,” I said. “Maybe he’s whispering it quietly.”
“Could be,” my dad said. “But maybe he doesn’t need to. Maybe the four cards work as a code, telling you the fifth.”
“But the professor doesn’t get to pick the cards,” I said.
“He does put them in order,” my dad said. “And maybe that’s enough. Maybe, given five cards, you can always sequence four of them into a code for the fifth.”
We grabbed a deck of cards from the cupboard and began working through the puzzle aloud. “We’ve got five arbitrary cards,” my dad said. “Four we’ll reveal, and one will remain hidden. First, we need to pick the hidden one wisely.”
“Which should we pick?” I said.
“Well, there are four suits in the deck—spades, hearts, diamonds, and clubs. So with five cards, there’s no way for them all to be from different suits. Among the five, you’re guaranteed to have at least two cards of the same suit.”
“Why does that help, to have two cards of the same suit?” I asked.
“Because,” he said. “We pick one of them to be the hidden card, and the other one we always sequence first.”
“Why?”
“That way, the first card in the sequence always spells out the suit of the hidden card. It’s our indicator card.”
“Okay,” I said. “But how do we know what number the hidden card is? It could be the 2, the Jack, the King, the 7, the 5…”
“Hmm.” My dad furrowed his brow. “Well, there are thirteen possible values: Ace, the numbers 2 through 10, Jack, Queen, and King. We can put them in a circle, like a clock.” He drew a picture:
“Then,” he continued, “the largest possible distance between a pair of cards is six steps.”
I nodded. “But how does that help us?”
“We’ll agree in advance to always take our steps clockwise,” he said. “It’s important that we agree on the direction in advance.”
“Okay,” I said.
“Now, I think of the indicator card as the starting point, and the hidden card as the endpoint. So I just need to communicate the message of how many steps to go—anywhere from 1 to 6.”
“But what about Ace to 8?” I asked. “That’s seven steps.”
“True,” my dad said. “So I would have to pick 8 as the indicator card, and Ace as the hidden card. That way, it’s still just six steps.”
“Hmm,” I said. “But how do you convey the number of steps? You’ve only got three cards left to send the message.”
“Luckily, there are exactly six ways to order three cards. If we label them ‘small,’ ‘medium,’ and ‘big,’ then I can put them in the following orders:
“You can even think of these as three-digit numbers,” he added. “The smallest possible three-digit number means, ‘add one,’ the largest means ‘add six,’ and the others have meanings in between.”
“But what if the sequence has two cards with the same value?” I asked. “Like two Jacks?”
“Then we’ll need to use the suits to distinguish,” my dad replied. “We’ll agree that Spades are highest, Hearts next, then Diamonds, and last are Clubs.”
“Hmm,” I said. This was a little much for me.
“We’ll practice,” my dad said. He picked five cards, quickly concealed one, and laid out the other four in careful order.
“Well, the hidden one is a diamond,” I said.
“Good. And what’s the number?”
“It’s…” I paused. “It’s Jack, plus one. So it’s the Queen of Diamonds.”
My dad flipped over the card. And so it was.
We practiced more. I looked at this spread of cards:
“It’s a spade. Plus six steps, so… the 8 of Spades,” I said. My dad grinned.
I looked at this foursome:
“5 of Hearts,” I declared.
“Focus your psychic powers,” my dad advised, and I saw my mistake.
“4 of Hearts,” I hastily corrected.
“Good work, my psychic son,” he smiled.
It became our little party trick. And now, with this post, I’m busting our illusion open. We had a good run. I hope you enjoy it as much as we did.
Great reading!
When we were in Florida we had this conversation, reversed engineering 🙂
Thanks! Yeah, I love intellectual reverse-engineering. A high school friend once warned me to destroy a paper airplane before our other friend reverse-engineered it.
This is a cool trick, with the mathematical economy of needing exactly 6 steps, and 3! being 6.
I was trying to figure out how to remember the 1-6 part, and came up with this: compare each of the 3 cards to its predecessor, and if the sequence goes
up-up it’s a 1.
down-down it’s a 6.
up-down it’s even (2 or 4)
down-up it’s odd (3 or 5).
Then,
if the first is smaller than the last, it’s 2 or 3
if the first is larger than the last, it’s 4 or 5.
Yeah, that seems like a reasonable method.
For me, it’s easiest to think of the sequences as ordered numbers. Say our three cards are 2, 5, and 7. If you think of the sequences as three-digit numbers, it’s easy to order them from smallest to largest:
257
275
527
572
725
752
And this corresponds exactly to the +1 to +6 information.
I like your way better now than I did when I first saw it. Say we have 527; I read that as “5 is the 2nd out of the 3 digits, so it’s in the second group of 3; 27 is lower than 72, so it’s the first of the second group; therefore #3.
Yeah, reflecting on my thought process, that “second group out of three; first member of the second group” thing is exactly what I do, too.
That is the coolest thing I’ve read today. I’ll need to bookmark this and practice it for use in the future!
Glad you enjoyed it! It’s a fun way to waste half an hour with a deck of cards.
A clever rouse.
Fitch Cheney was evidently a clever fellow.
Here’s another concept to remember the correct sequence.
Assign each of them a number:
Big=1
Medium=2
Small=3
Start with 3
Add the “jump” between the first and second card
Add 1 if that first jump was positive
Add the “jump” from the second to the third card
Example:
Medium, Small, Big -> 231
1st jump is [+1], add [+1] because jump was positive, 2nd jump is [-2]
Result: 3 + 1 + 1 – 2 = 3
Big, Medium, Small -> 123
Result: 3 + 1 + 1 + 1 = 6
Interesting. A cool trick in itself!
A simpler design:
Choose the four cards so that two are red and two are black. The highest-ranked card encodes the suit of the hidden card. The other card of the same color is ignored. The two remaining cards are summed to give n+1, where n is the rank of the fifth card. In this way, you can even allow the victim to shuffle the four cards, since their order contains no information.
but it’s possible that all 4 or all 5 cards are the same colour…
I think that’s a different trick (since in this set-up, you don’t get to pick the five cards, just their sequencing), but a nice one, too.
From what I gathered, you aren’t actually choosing the 5 cards from the deck. Instead you’re choosing 1 card to hide and arranging the other 4 randomly drawn cards to convey the identity of the hidden card.
http://thephantomtollbooth.wikia.com/wiki/The_Mathemagician
Man, I need to go back and read that again.
This is a great trick! Now expand it from the 52-card deck; what is the biggest size deck you can do this 5-card trick with (still with 4 suits)? what about if you added another card to the code? Is there a general form for a m-sized code given an n-sized deck?
There exists a code that works for a 124-card deck. In general, when you draw n cards and hide k of them, there is a scheme that works for a deck of (n! / k!) + (n – k) cards. I leave it as a challenge to design a set of 124 cards and a system that makes it easy for humans to use.
Hmm this has me stumped. I can manage the simple n=2, k=1 case with 3 cards but even the next case n=3, k=1 with 8 cards has me stumped. I’d love some further details Nevin.
Nice! I’ll have to check out this 124-card version and the m-n-generalization…
You need to order the suits so that even if you have twice the same value you can still apply the big/medium/small ordering…
True. Depending who you talk to, Spade-Heart-Diamond-Club and Spade-Heart-Club-Diamond are both reasonable orderings, so you’d have to specify in advance.