Welcome to The STEM Sessions Podcast. I am Jarl Cody, your host and narrator.
Several years ago, I attended a Penn & Teller show at which they performed their “Card Trick to Find Love”. You can find several performances of the trick on YouTube, but the quick summary is audience members rip four random playing cards in half, stack them, perform a series of shuffles as instructed by Penn, place the top card against their heart, perform another series of shuffles and throwaways, and if the last card remaining is the matching half of the card against their heart, then they will find love.
Penn then asks the audience to cheer and hold up their cards if they found love, while he and Teller search the audience for someone whose cards do not match. After finding said person, Penn stammers with excuses for why the trick didn’t work, before shouting “…or it’s because you’re too stupid to follow simple directions” confirming the trick is actually a trick.
Well, me being me, in the following days, I made a spreadsheet showing the card arrangement after each step in the trick, and deconstructed the math. While I will leave that task for you to repeat on your own time, I’ll discuss a few of the steps now.
This is The STEM Sessions Podcast – Episode Eight. Algebraic Performance
A common trick employed my magicians and psychics is asking their audience to pick a number, then instructing them to perform a series of mathematical operations, deriving a new number, which the performer correctly guesses. Of course, we know this isn’t magic or clairvoyance. We know the performer forces the results via the math, whether or not we understand the details.
In magic, a “force” is the action taken by a magician to get you to do what he needs you to do for the trick to be successful. It’s the building block of nearly all tricks with audience participation. The force could be ensuring you select the predetermined object at the beginning of the trick, or making you perform a series of operations that will always produce the same end result regardless of the starting number.
Let’s run through a very rudimentary example as you might hear it live:
Pick a number, any number. Do you have a number? Good. Now double it. Now add ten to that new number. Everybody still with me? Fantastic! Now I want you to divide the sum by two, and then subtract your original number. You all now have a new number in your head, and now I’m going to guess what it is. Is it five?
When the instructions are given verbally, in a crowd, with excitement and a magician’s flare behind them, it’s easy to get caught up in the trick; especially when you’re focused on correctly implementing the current step and listening for the next one so you won’t fall behind. But when written out algebraically, it’s easier to see what’s occurring.
Let’s run thru the math in the instructions.
Pick a number: N
Double it: 2N
Add ten: 2N + 10
Divide by two: (2N + 10) / 2 = N + 5
Subtract the original number: N + 5 – N = 5
The magician then asks if your number is five. He’s correct. And everyone is amazed.
When viewed in this format, the force is obvious. Regardless of the starting number, the final result will always be five, because, in the end, the series of operations is designed to cancel out the original number and leave you with the number five.
What’s more, there is nothing overly complex about the above series of operations. In fact, given a bit of time, one can develop any number of combinations that will force a desired answer. Here’s another example:
Pick a number: N
Add six: N + 6
Multiply by three: (N + 6) x 3 = 3N + 18
Subtract 18: 3N + 18 – 18 = 3N
Divide by the original number: 3N / N = 3
The magician guesses three. He’s once again correct. And again, everyone is amazed.
After seeing a few of these, you start to understand the methodology. Later operations undo prior operations in the guise of adding complexity, even though they’re actually simplifying the problem. And at some point, the original number needs to be cancelled out.
Similar manipulations are the secret behind many card tricks. In Penn & Teller’s “Card Trick to Find Love”, the audience member starts with four unique playing cards (no pairs), selected by the audience member from tables of cards before the show. When the trick starts, the audience is asked to place their cards in any order, face down, rip the stack in half, and place one stack on top of the other, resulting in a stack of eight half-cards.
From the audience perspective, they have a stack of random cards arranged in a random order. But from Penn & Teller’s perspective, they have a stack of random cards in a known order. Let’s call the top card position 1 and the bottom card position 8. The first card has its halves in positions 1 and 5; the second in positions 2 and 6; the third in three and seven; and the fourth in four and eight. The suit and rank are irrelevant.
The first three shuffles are designed to force the halves of the original third card into positions 1 and 8; the top and bottom of the stack. Even the step in which you’re given the choice of placing the top three cards anywhere in the middle of the deck forces the desired arrangement, Because, despite the choice given to the audience, the final location of those three cards is irrelevant as long as they’re placed between the top and bottom cards.
The half of the original third card in position 1 is the card you instructed to hold to your heart. This means, no matter what happens in the rest of the trick, the second half of that card, currently at the bottom of the stack, must always be in a safe position, because it needs to be the last card you’re left with.
And that’s exactly what happens. Whether you choose to throw away one or two or three cards in the next step, the matching half will always be safe. If you throw away one or three cards, the matching half will be at the bottom position in an even number of cards. If you throw away two cards, the matching half will be at the bottom position with an odd number of cards in the stack.
This arrangement is important, because it means the next series of seven shuffles always puts the matching half in an odd numbered position to start the Loves Me Loves Me Not elimination round. If you’re like me, you diagramed these three cases, because it’s the only way to conceptualize the movement of the cards. Or, if you’re better at math than me, you would know to use modular arithmetic to show the results.
To understand modular arithmetic, think of a clock. If you add five hours to 9:00A, you arrive at 1400 or 2:00P in a mod 12 system. It’s a mod 12 system, because there are 12 hours or 12 discrete locations, and you restart the counting after 12. So 9 + 5 in a mod 12 system would be (9 + 5) – 12, or 2.
In the case where one card was thrown away, the matching half is in position 6 of a six card stack or a mod 6 system. During the next series of shuffles, the card moves up the stack seven places. Using modular math, that is (6 – 7) + 6 which gets you to position 5. If you removed two cards, you have a mod 5 system. If you removed three cards, you have a mod 4 system. Similar calculations show the final positions to be Position 3 in the second case, and Position 1 in the third case. All odd numbered positions. And those odd numbered positions mean the matching half of the card doesn’t get cast aside by the “Loves Me Not” throwaways, which are the even steps in the process.
Granted, that’s an over-simplification, because which odd position and how many total cards are also critical components of the math involved. But the point is, from the very beginning, the result is a forgone conclusion. Any choice you’re given through out the trick, ends up being irrelevant to the positions of the critical cards, and only adds the illusion of your influence to the performance.
And that’s the real trick – the performance. It entertains you. It distracts you. Even if you made up your mind ahead of time to observe the trick critically, a good performance will overwhelm your sense of skepticism and suck you in. The performance Penn & Teller give is a great example of this. Their entire philosophy is they want you to know you’re being played; that the true magic is the skill of the performer and his ability to make you suspend your disbelief. They dare you to follow the math and reverse engineer the mechanics, but you still get lost in their performance. And yet, despite being fooled, I think you emerge the better for it in the end.
Thank you for listening to The STEM Sessions Podcast.
This episode was researched, written, and produced by me, Jarl Cody.
While I strive for completeness and accuracy, I encourage you to do your own research on the topic we discussed, and confirm what I’ve presented. Corrections and additional information are always welcome.
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Finally, please remember STEM is not the exclusive tool of experts, policy makers, or talking heads. Every presenter is susceptible to unconscious and, sometimes, deliberate bias, so always verify what you read and what you’re told.
Until the next one, stay critical.
Mathematical Explorations of Card Tricks
Timothy R. Weeks