The magic of maths

Summary

Podcast Summary: "The Magic of Maths"

Podcast: More or Less
Host: Tim Harford
Guest: Professor Percy Diaconis (Stanford University, mathematician and magician)
Date: August 17, 2013

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Overview

In this engaging episode of More or Less, Tim Harford explores the fascinating connection between mathematics and magic. The centerpiece is an interview with Professor Percy Diaconis, a renowned mathematician and professional magician, who unpacks the elegant mathematics behind classic card tricks. The episode blends magical demonstrations with deep dives into mathematical principles, showing how mathematical thinking both inspires and is inspired by the world of magic tricks.

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Key Discussion Points & Insights

1. Demonstrating Mathematical Magic (00:23–02:29)

• Live Trick Performance: The episode opens with BBC editor (and magician) Hugh Levinson performing a classic card trick originated by Diaconis. After Tim repeatedly shuffles, flips, and splits a selection of cards, the five face-up cards reveal themselves as a perfect royal flush.
• No Sleight of Hand: Levinson emphasizes that “there was no sleight of hand… somehow it works” (02:35), highlighting the baffling effect of the trick even to experienced performers.

Notable Quote

“That is completely extraordinary. That’s amazing. Are you going to tell me how you did that?”
— Tim Harford (02:29)

2. The Mathematics Behind the Magic (02:47–04:35)

• Professor Diaconis Explains: Diaconis joins the show and reveals the underlying math in seemingly magical card arrangements. By using simple operations—flipping cards, cutting the deck—certain properties (invariants) of the deck are preserved, making the trick possible.
• Hands-On Illustrations: He instructs Tim (and listeners at home) to try with four cards, demonstrating an invariant arrangement that persists, regardless of how the cards are cut.
• Elegant Simplicity:

  “There’s an invariant that stays put.”
  — Professor Percy Diaconis (03:46)

Key Understanding
Mathematical invariants—properties that remain unchanged through certain operations—are the foundation of this card magic.

3. The Life of a Magician—and a Mathematician (04:40–07:31)

• Unconventional Path: Diaconis shares his remarkable story: running away from home at 14 to travel with legendary magician Dai Vernon, becoming immersed in the world of professional magic before discovering mathematics through curiosity about probability.
• Discovery of Probability: His fascination with questions like dice probabilities led him to study William Feller’s book on probability, which, while inaccessible at first, planted the seeds for his future academic pursuits.
• Unusual College Journey: He eventually enrolled in night school at age 24, was rejected by the math department at Harvard due to his unconventional background, but was accepted by the statistics department with help from a letter by noted math and magic writer Martin Gardner.
• Persistence:

  “Of the 10 best card tricks invented in the last 10 years, this kid invented two of them. Maybe you should give him a chance. They let me in.”
  — Professor Percy Diaconis (06:58)

4. The Interplay Between Magic and Mathematical Research (07:31–08:53)

• Magical Problems Inspire Math: Diaconis discusses how the requirements of magic tricks spark serious mathematical questions, and vice versa.
• Perfect Shuffles: He describes a seemingly simple yet mathematically complex property of shuffling: performing eight perfect shuffles on a 52-card deck returns it to its original order—but the corresponding questions for decks of different sizes can be extraordinarily challenging.
• Enduring Mysteries:

  “There are things that a 13 year old kid could think about that touch mathematics in a very deep way and are beyond modern mathematics to understand.”
  — Professor Percy Diaconis (08:44)

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Notable Quotes & Memorable Moments

• Tim’s Astonished Reaction:

  “That is completely extraordinary. That's amazing.”
  (02:29)

• Diaconis on Invariants:

  “There’s an invariant that stays put.”
  (03:46)

• On Unsolved Mathematical Problems:

  “That question is beyond current mathematical understanding.”
  (08:32)

• On the Deep Connections Between Magic and Maths:

  “Magical effects can inspire serious mathematical research. And vice versa.”
  (07:44)

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Timestamps for Key Segments

• 00:23–02:29: Live card trick performance and reveal
• 02:47–04:35: Diaconis explains the mathematics of card invariants
• 04:40–07:31: Diaconis’ unconventional journey from magic to mathematics
• 07:31–08:53: How magic and mathematics inspire each other; perfect shuffles and unsolved questions

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Summary

The Magic of Maths offers a delightful intersection of magic and mathematics. Through live demonstrations and insightful conversation, the episode shows how simple pleasures—like a baffling card trick—are deeply rooted in elegant mathematical ideas. Professor Percy Diaconis’ personal story and expertise bridge these worlds, inspiring listeners to see magic in mathematics, and mathematics in magic.

Transcript

[BBC Announcer]

Thank you for downloading from the BBC. For details of our complete range of podcasts and our terms of use, go to bbcworldservice.com podcasts

[Tim Harford]

hello and welcome to More or Less on the BBC World Service. We're your guide to the numbers that surround us in the news and in life. And I'm Tim Harford.

[Professor Percy Diaconis]

It's a trick that I've studied and developed. Hidden behind it is some rather elegant mathematics. It's not so hard.

[Tim Harford]

We're honoured to have a guest appearance from a top professor of maths and statistics. Percy Diaconis is by all accounts, a brilliant mathematician and a brilliant magician. Before we speak to the great man himself, I've booked another special treat. Someone who can demonstrate one of Percy's tricks right in front of my eyes. The editor of Crossing Continents, Hugh Levinson, doubles as the BBC's in house magician.

[Hugh Levinson]

Now, you've seen these cards, they're completely normal cards. I'd like you to take off. I'd know about a third of the deck, something like that.

[Tim Harford]

Okay, so I've got 20 or so cards in my hand.

[Hugh Levinson]

Okay? What we're going to do is we're going to mix them in six different ways, okay? And what I'd like you to do is to shuffle overhand.

[Tim Harford]

So just the regular, regular shuffle, incredibly competent, normal shuffle. Okay? Right, so I've done that.

[Hugh Levinson]

Right. Okay. I'm going through the cards and some of them I'm turning face down. Now, here's a pair of cards. You could have them this way up or the other way up. It's up to you, Tim.

[Tim Harford]

Okay, so I can see an 8 of spades face up and there's another card face down. Flip them the other way round.

[Hugh Levinson]

You did what you like. Now, deal the cards into two piles, alternately right and left.

[Tim Harford]

So just left, right, left, right, left, right, left, right, left, right, left. Some of them are face up, some of them are face down. Okay, now, two piles in front of us. Right?

[Hugh Levinson]

Choose either pile. Whichever pile you choose, I want you to pick up, turn over and put on the other pile. Your choice.

[Tim Harford]

Ok, so I'm going to pick up the left hand pile, turn it over and place it on top of the right hand pile.

[Hugh Levinson]

Okay. And I can see by the way you've handled these cards that actually you're a card sheet, a poker player, I'd say, because when we spread the cards out, some are face up, some are face down, as you'd expect with this kind of trick. But if we look at the five cards that are face up. Those five cards, after all this shuffling,

[Tim Harford]

are the Jack, Queen, Ace, 10 and King of spades. So all the royal.

[Hugh Levinson]

The royal flashing spades. The highest possible hand in poker.

[Tim Harford]

That is completely extraordinary. That's amazing. Are you going to tell me how you did that?

[Hugh Levinson]

What I can tell you is there was no sleight of hand. And the key bit is that bit with the pairs of cards where they go face up and face down. Now, I have read about this. I still don't understand it. It baffles me that it works, but somehow it works.

[Tim Harford]

What we really need is somebody who is both a professional mathematician and a professional magician. And by the magic of radio, we have such a person, Professor Percy Diaconis, who joins us from Stanford University. Professor Diaconis, do you know how this trick works?

[Professor Percy Diaconis]

Well, yes, I do. It's a trick that studied and developed. In fact, hidden behind it is some rather elegant mathematics. It's not so hard if you have four cards there or if people who are listening at home can find four playing cards.

[Tim Harford]

I have four playing cards now.

[Professor Percy Diaconis]

Okay. I'd like you to turn them in the following way. They should all be face down.

[Tim Harford]

Yep.

[Professor Percy Diaconis]

And now if you can, turn over every other card.

[Tim Harford]

Okay.

[Professor Percy Diaconis]

And now hold them in your hand as if you were gonna deal or something like that.

[Tim Harford]

Yep.

[Professor Percy Diaconis]

If you cut one card from top to bottom.

[Tim Harford]

Yep.

[Professor Percy Diaconis]

How are they?

[Tim Harford]

Okay, they're face down, face up, face down, face up.

[Professor Percy Diaconis]

Right. And you could keep doing that. That is, you could cut any number of cards from top to bottom, two or three, one. And they would be alternating. They would either be down, up, down, up, or up, down, up, down. So that's our first idea, that there's an invariant that stays put.

[Tim Harford]

When I think back to what Hugh did to me, I'm thinking, well, he did, at one stage, just go through the deck, looking at the deck, and he flipped some cards over. So presumably at that point, he was able to set up his royal flush in a way that was not at all transparent to me and then give me multiple opportunities to do what appeared to be a full shuffle, but in fact, wasn't really making any difference to the fact that Q had put this oil flush in there. Is that. Have I got it?

[Professor Percy Diaconis]

If you were being graded on that answer, you'd get an A in my class.

[Tim Harford]

Listeners who don't know your story will be imagining this nerdy kid, probably very good at maths at school, maybe a little sideline in card tricks, and then goes on to university and college, and then eventually becomes a math professor and does math tricks on the side. But actually your road to becoming a mathematics professor and becoming a professional magician was a little bit more chaotic than that.

[Professor Percy Diaconis]

That's, that's certainly true. I grew up in New York City and the kids in New York City used to congregate at a local magic shop on Saturday afternoons. And I met the greatest exponent of purest sleight of hand of the 20th century, Dave Vernon, his name is. And one day he called me up and said, you know, I'm going to Delaware tomorrow, do you want to come? Going to a magic convention. And I didn't think twice. I packed a bag and. How old were you? 14.

[Tim Harford]

Home with a card shard, ran away from home.

[Professor Percy Diaconis]

And we began traveling about and it was a wonderful, very supportive, protective world of inner circle magic. And we would go from city to city and he would have me demonstrate things and teach me tricks. And I literally never went back. I saw my parents about 17 years later. That was unusual. And now I, of course have kids of my own. If one of them disappeared, I would go crazy. But somehow the police had other things to do than search for 14 year old runaway magicians. And I was left alone to traipse about and got interested in mathematics through magic in the following way. I heard a lot of talk about probability and seemed to have a little bit of a gift for thinking about, oh, suppose you roll a pair of dice, is it more likely that you get two sevens or a six or an eight? I asked a friend, is there someplace to read about this? And he pointed me to a book by William An Introduction to Probability and its Applications. And I bought a copy when I was 16 and I couldn't read it because I didn't know calculus, but I could turn the pages and I could see it was just fascinating.

[Tim Harford]

You ended up at Stanford University as a professor, so presumably at some stage you took a.

[Professor Percy Diaconis]

It's true. When I was 24, I enrolled at night in City College in New York's big free school. And my teachers refused to write letters to Harvard because they said, nobody from City College has ever gotten into the Harvard math department. So I applied to the statistics department instead. I didn't actually know what statistics was about. And I got Martin Gardner, who was a magic friend, to write a letter. And he wrote a letter saying, of the 10 best card tricks invented in the last 10 years, this kid invented two of them. Maybe you should give him a chance. They let me in.

[Tim Harford]

I'm intrigued. In your work, do you find the Magical effects inspiring serious mathematical research. Or is it the serious mathematical research that is inspiring the magical effects? Or both?

[Professor Percy Diaconis]

It does go back and forth. For example, one of the things I learned as a youth is that you can cut a deck of cards in half perfectly and shuffle them, dropping them 1, 1, 1, 1, 1,. So that every other card drops. And it's what would be called a perfect shuffle. Since it's perfect, there's no randomness to it at all. One of the things you find is that if you repeat that eight times with a 52 card deck, the cards come back to order and it's, it's quite astounding. And now if you try to think, well, suppose I shuffle a 54 card deck, how many shuffles does it take to recycle? The answer changes. It takes 52 shuffles to recycle a 54 card deck. Well, if you ask, suppose I had a deck of general size, two end cards, how many times do you have to shuffle it to have it recycle? That question is beyond current mathematical understanding. And I find it wonderful that there are things that a 13 year old kid could think about that touch mathematics in a very deep way and are beyond modern mathematics to understand.

[Tim Harford]

Professor Percy Diaconis, the author of Magical the Mathematical Ideas that Animate Great Magic Tricks.

That's all we have time for this week.

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