Category: Mathemagical Thinking

Am I Being Random?

We’re going to be talking about patterns.

Look at the following three sequences of playing cards.

Sequence 1 Sequence 2 Sequence 3
7 ♣ J♠ 8♠
10♡ K♣ A♡
K♠ 5♣ A♢
3♢ 2♡ 7♡
6♣ 9♠ 3♣
9♡ A♠ 7♣
Q♠ 3♡ 9♣
2♢ 6♣ 2♡
5♣ 8♢ Q♠
… … …

What can you tell us about each of the three columns? Start with the one on the left.

A few seconds spent looking at the first column and you will notice a black-white-black-white pattern repeating. On closer inspection you might see Club-Heart-Spade-Diamond repeated. Finally, if you are really paying attention, you’d see that the numbers increase by three each time: 7 to 10 to 13 (King) to 3 to 6 to 9 etc. We have here a simple pattern, and it gives us a stacked deck.

Stacked decks are very useful to the magician in many circumstances. For example, suppose you are asked to pick a card and the magician peaks at the one next to it then he immediately knows which is yours. Or if you take one out and put it back elsewhere he will know which is the one out of sequence, and that’s the one you picked. Those are two easy, beginner, tricks. More sophisticated effects use the fact that the magician knows the position of each card in the deck.  

The particular stack in the first column goes by the name “Si Stebbins,” the stage name of William Coffrin. Although there is a pretty clear pattern here — the suit order is easily remembered by the mnemonic CHaSeD — believe us when we say that with suitable distractions the layperson is unlikely to spot this when the magician briefly waves the deck in front of them. But sometimes the magician needs a stack that can stand up to closer scrutiny…

The second column is also part of a stacked deck, this time the Aronson stack, invented by magician, mentalist and lawyer, Simon Aronson. The stack is designed to look random but also once you’ve memorized this stack there are many tricks that work simply because of the clever order of the cards.

The third column is genuinely random, in the sense that we shuffled a deck and wrote down the first nine cards.

Being able to see patterns is important.

However humans also tend to see patterns even when there’s no pattern to be seen. The tendency to see patterns in random data is known as apophenia. We see shapes of animals in clouds. Or images of Jesus in a slice of toast. In finance the technical analysts promote the idea of trendlines and patterns in stock-market graphs, even though the proper statistics tell us that such lines and patterns have no predictive power. In extreme cases apophenia can be an indicator of delusional thinking or schizophrenia.

The magician knows about apophenia and so his goal with a stacked deck is to have it look random to the casual observer (or even to someone who “burns” the cards, i.e. stares intently at them). But there’s a big difference between being random and looking random, given the human urge to see patterns.

Let’s move on from cards to coins, and look at another sequence. Toss a coin many times, write down “H” when it lands Heads up, and “T” if Tails up. Here’s an example:

HTTTTHTTTTHHTHHH…

That’s a nice random sequence, no?

No! One H, four Ts, one H, four Ts, two Hs, one T, three Hs,… = 1.414213… The first digits in the square root of two.

How about this one:

THTTTTTTH…

You must be onto us by now. That’s a lot of Ts in a row, must be the part of some pattern. No! We generated that in Excel using one of its random number generators and it was literally the first sequence we did. We see patterns when there aren’t any and miss patterns when there are.

Exercise: Try writing down a sequence of Hs and Ts and try to make it look random. You’ll probably be reluctant to put down six Hs or Ts in a row. It just wouldn’t look sufficiently random.

Swingers

This post looks at Mathematical Thinking, which we define as “The belief, especially characteristic of scientists, that events can be understood, described, and predicted using mathematical equations; thinking founded on this belief.” Sometimes it’s a good thing, sometimes not so good.

Imagine you’re at the bar, and order a gin martini. The bar man serves it in the usual conical martini glass. Tell us, when the drink is halfway down the side of the glass how much is left?

Half? No, the glass is pointy at the bottom so there’s clearly less in the bottom half than in the top half. A quarter? Final answer. One quarter!

Wrong.

You may recall from school the idea of similar triangles. They are triangles with the same shape, but a different size. This is what we also see in the cone, our martini glass. The bottom half of the martini glass is “similar” to the full glass.

The problem of calculating volumes of geometric objects goes back to the ancient Greeks. Draw a square on a piece of paper and then draw another with sides twice the length. How much has the area increased by? It’s four times bigger. You can see that four of the smaller squares will fit inside the larger. Now take eight sugar cubes or dice. Put four into a square and put another four on top. You’ve now got a larger cube, each side of which is twice the length of the smaller. So for a cube halve the length, you’ve got a volume that’s a mere eighth of the original!

The shape doesn’t matter here. All that matters is that the half-full glass is “similar” in the mathematical sense to the full glass. The formula is just 1 divided by 2 raised to the power d, where d is the number of dimensions. Since we live in d = 3 dimensions – we’re leaving out spooky extra dimensions – the answer is 1 divided by 2 cubed, which is 1/8, or 12.5%.

Or in mathematical terms: it’s time to order another.

Mathematicians are full of tricks like this where pure thought can lead to answers that are not otherwise obvious or where the brain gets tricked. Say that, reaching for a bottle on an upper shelf behind the bar, the barman’s arm brushes against a light hanging from a cable, making it swing gently. What is the time taken to swing left to right then back again (i.e. the period)?

Galileo first became interested in this problem when he noticed the sway of a chandelier in the Pisa cathedral. There are a few ways of solving it mathematically. Most require some knowledge of physical laws, but there is a simple method known as dimensional analysis that gets you most of the way there. As the name suggests, it is again based on the concept of dimension.

We first ask what quantities could affect the period of the swing. A couple are obvious: the mass of the weight, and length of the string. Anything else? The colour of the string or the time of day shouldn’t come into it, but one can imagine the answer is different on the moon from here in the bar so gravity must play a role. The acceleration due to gravity, usually represented by g, is 9.8 meters per second squared. On the moon it is only about 16 percent of that number, because the moon is less massive.

Now let’s postulate that the period is given by multiplying or dividing those numbers together, where again you are allowed to raise a number to a power. We will include fractional powers, so for example a number raised to the power ½ is the square root. The trick is that, for the formula to work, the units of the expression of the period, which are seconds, have to be the same as the units of the expression that you just made up. The two sides of the equation must be made of the same stuff.

The two sides of an equation must be made of the same stuff.

Write it out and you find that the only way to make it work is if – ta da! – the period of the swing is proportional to the square root of length divided by the gravitational constant g. So the mass doesn’t – and can’t – come into it.

There is no magic involved here, no spooky automatic writing or communication with the other side, but in a sense the result is magical, because it shows us something very counterintuitive, namely that the time of the swing doesn’t depend on the mass of the pendulum. The weight only comes in through the acceleration due to gravity. On the moon gravity is less and the swing will take longer. In fact, you can use this relationship to test tiny local perturbations in the gravitational field.

It turns out that if the length is exactly one meter, then the period T is almost exactly 2 seconds, and a single swing counts out a second. Which is why traditional grandfather clocks, first produced in the late seventeenth century, have a pendulum of this length. It also inspired the definition of the metre a century later.

This technique of dimensional analysis has been used to great effect in other circumstances. New Mexico was the site of the first explosion of an atomic bomb, at 5:29am on July 16, 1945. Codenamed “Trinity,” much of the project – including the amount of energy released by the explosion – was obviously kept secret. However one observer at the explosion was the mathematician GI Taylor. Taylor is famous for his work in fluid and solid mechanics, in particular the difficult problem of turbulent flow. He estimated the energy in the atomic explosion by applying precisely the above dimensional trick but with quantities energy, air density, radius of the explosion, and time. He estimated the energy released to be 17 kilotons of TNT. President Truman later revealed it to have been 22 kilotons. Taylor’s approximation was remarkably accurate and shows the power of quite simple mathematics, or perhaps more precisely the power of Mathematical Thinking. As we’ll see, similar thinking was behind the bomb itself – this is a dangerous kind of power.

Few people are as clever as GI Taylor. And unfortunately there are many people as gullible as Doyle. But there is also an intersection between magical thinking and mathematical thinking where the power and beauty of mathematics gets used as a decoy to confound while giving the illusion of clarity. We first discovered this happening in finance and economics. But once you’ve been alerted to the possibilities of such abuse you start seeing examples everywhere.

This is what we call Mathemagical Thinking. And in the wrong hands it can be as dangerous as nuclear weapons.

Magos

Magic and science have a surprising amount in common. They are both about telling a story, and constructing a narrative of events. In one, the story appears magical, in the other it appears mathematical, but that difference is less important than might appear. They can both be dangerous if misapplied. In fact, one could argue that science is truer to the spirit of magic than most modern magicians are.

Back in ancient Greece, where the term originates, the magos (magicians) were religious figures, like shamans, who were believed to have access to the Otherworld. They performed rituals and ceremonies, administered healing potions, cast spells on enemies, or contacted the dead through seances. They were not usually connected to official temples – who viewed them with suspicion – but operated as freelancers, selling their services in the private market.

Such gnostics and mystics have throughout history operated on the fringes of mainstream religion – with “superstitions” instead of official sacraments, and “witchcraft” instead of sacred rites. But they still had considerable knowledge and power, and were far more than a source of entertainment. They didn’t do children’s parties. And the separation between magic and science was not so clear.

Pythagoras, who is sometimes described as the first pure mathematician, ran what amounted to a pseudo-religious cult with all kinds of strange teachings and an interest in esoteric symbols. The first mathematical models of the cosmos were developed by Greek mathematicians expressly for the purpose of reading the future through astrology. Chemistry grew out of alchemy, whose practitioners included scientists such as Isaac Newton. In the late nineteenth century, the discovery of the cathode ray tube, with its eery green glow, seemed to excite the spiritualist community as much as it did the scientific community (though that changed when it became the basis for TVs).

Even today, science is often as much about putting on a great show and amazing us with magical demonstrations as it is about making life-altering discoveries. Consider for example the moon landing of 1969 (though some claim it was a staged illusion) which was actually about impressing the Russians with military technology. Or more recently the excitement about the discovery of a tiny ghostlike particle known as the Higgs boson, which sounds like something a wizard would concoct.

Perhaps the main difference now is that magicians are seen as mere paid entertainers, while scientists are revealing the deep truths of the universe. Our aim in this blog is to break down this barrier to show the still-powerful connection between magic and science. Through a sequence of stories, historical nuggets, magic tricks, and other amusements, we reveal some of the tricks that scientists play on their audience. And we will show how even in the modern scientific age, many people are prone to mathemagical thinking.

Automatic Writing

On the topic of gullibility, here is a story that we like to share with our banker friends during our “seminars” on magical thinking (much better paid than real magic shows and the audience is less demanding).

In the summer of 1922, Sir Arthur Conan Doyle invited his friend, the magician Harry Houdini, and his wife Bess to join them for a week at Atlantic City. The vacation was going well, and everyone was having a good time, until Sir Arthur suggested that his wife Jean could put Houdini in touch with Houdini’s beloved mother Cecelia, who had died almost a decade earlier.

Sir Arthur was of course famous for being the creator of the ultimate scientific rationalist Sherlock Holmes, however he had also been converted to spiritualism after losing his son and brother during World War I. When he saw Houdini perform his act during a tour through England, he became convinced that Houdini had genuine supernatural powers. And he had great faith in his wife’s talent as a medium.

For Houdini, the situation was complicated. For one thing, he knew that his illusions were the product of stagecraft, not magic, and that Sir Arthur was overly gullible. He had also authored a book in which he revealed the tricks of a number of magicians and mediums. It was inspired in part by earlier, unsuccessful attempts to contact his mother, who he had seen as “the guiding beacon of my life.” But at the same time, he liked Doyle – and he really missed his mother. So he agreed to give it a go.

The séance was held at the Doyles’ hotel room. After turning down the lights, Lady Jean went into what appeared to be a deep trance. Then she grabbed a pen and started scribbling manicly on sheets of paper – what spiritualists called automatic writing – as if she were channeling the spirit of Houdini’s mother. Houdini read the sheets as she finished them, fifteen in all before she wound herself down. But his awkward feelings were not relieved.

The letter said all the usual things that one might expect from a long-dead mother (love you, missing you, the Doyles are great, etc.) but it was written in perfect English (his mother was Hungarian and spoke little English), the first sheet was decorated with a cross (she was a Jew), she didn’t mention that it was her birthday that day, and so on. He didn’t need to be Sherlock Holmes to realise he was being conned.

Not wanting to spoil the evening, or their relationship, he didn’t say anything at the time; but matters came to a head later that year when he wrote an article saying that no medium had ever been proven to be able to contact the dead. Evidence-based, they were not. Doyle took it personally, and the friendship was effectively over.

Doyle continued to write books and give lectures promoting spiritualism. Houdini remained a skeptic, but still held out some hope that it might be possible to communicate with the departed – being a magician didn’t make him completely immune to magical thinking. He arranged a code with his wife, so that if she tried to contact him after his death through a medium she would know whether it was genuine. After he died in 1926, of a ruptured appendix on Halloween, Bess kept trying, but to no avail. She finally stopped the séances in 1936, famously announcing that “ten years is long enough to wait for any man.”

By that time, the craze for spiritualism was already losing some of its energy. And today, of course, most people see séances as something from an earlier, less scientific age. We know that for many people the desire to communicate with lost relatives is so great that they are willing to suspend disbelief; and that doing so makes them susceptible to the con-artists and fraudsters who Houdini had worked to debunk. But that doesn’t mean that we are immune to the power of magic – even when we are trying to be rational.

Bankers (and others) beware.

Magic Is A Sufficiently Advanced Technology

I have a saying, it is “Magic is indistinguishable from a sufficiently advanced technology,” which is very close to something Arthur C. Clarke said which is “Any sufficiently advanced technology is indistinguishable from magic.” Ok, I stole his saying and reversed it, I confess.

Sadly it is true. In olden times magicians would make balls levitate and float across audiences. It was a trick involving wires and threads. But you knew that. Now you can buy drones that do the same thing. Look at airhogs.com. Where is the magic in that?

In those good old days we read an audience member’s secret writing hidden inside an envelope using skills perfected over decades. (Often involving peaking. Actually always involving peaking.) Now you can buy writing pads that will transmit any drawing to a video screen on your wrist.

Cold reading an audience required psychological insights, and cojones of steel. “Is there anyone here called Margaret?” (I suspect Margaret is a very common name for an audience member at this sort of event.) But now I can see the names of people paying by credit card. I look at them on Facebook. Now I can say “Is there anyone here called Margaret, who went on holiday to Ibiza with friends and kissed a Spanish waiter?” Too easy. There’s no skill anymore.

What do you want, Instagram or magic? You can’t have both.