Alex Stone is the author of Fooling Houdini: Magicians, Mentalists, Math Geeks and the Hidden Powers of the Mind. His writing has appeared in DISCOVER, Harper’s, Science, The New York Times, and The Wall Street Journal.
There was a time when people thought of playing cards as cosmic instruments. Fortunes were told, fortunes were lost, and the secrets of the universe unveiled themselves at the turn of a card. These days we know better. And yet, a look at the mathematics of card shuffling reveals some startling insights.
Consider, for instance, the perfect, or “faro” shuffle—whereby the cards are divided exactly in half (top and bottom) and then interleaved so that they alternate exactly. Most people think shuffling tends to mix up a deck of cards, and usually that’s true, because a typical shuffle is sloppy. But a perfect shuffle isn’t random at all. Eight consecutive perfect shuffles will bring a 52-card deck back to its original order, with every card in the pack having cycled through a series of predictable permutations back to its starting place. This holds true for any deck, regardless of its size, although eight isn’t always the magic number. If you have 25 cards, it takes 20 shuffles, whereas for 32 cards it only takes 5; for 53 cards, 52 shuffles are needed. You can derive a formula for the relationship between the number of cards in the deck and the number of faro shuffles in one full cycle.
Amir D. Aczel has been closely associated with CERN and particle physics for a number of years and often consults on statistical issues relating to physics. He is also the author of 18 popular books on mathematics and science, and has been awarded both Guggenheim Foundation and Sloan Foundation fellowships. Many thanks to Steven Weinberg of the University of Texas at Austin and to Barton Zwiebach of MIT for their helpful comments.
Readers of this blog have probably heard the standard fare about how the Higgs boson “gives mass” to everything in the universe, probably with some kind of analogy, like the one about a famous person walking through a crowded room, pulled every which way by admiring crowds, and that these connections “make the person massive“—as the Higgs field does with particles. Now that we finally seemed to have pinned down the elusive particle, I want to explain where the Higgs came from and what it does. While our understanding of the particle comes from some complicated math, the formulas actually tell a fascinating story, which I’ll recount in this post. All you need to keep in mind is that in the modern understanding of physics, categories aren’t as starkly separate as you might think: particles can be represented as waves or fields, and a force can also be viewed as a particle or a field.
So, a fraction of a second after the Big Bang, the universe had four kinds of “photons” floating around—the usual photon of light, and three other massless particles that “look” and act just like the photon. We label them: W+, W-, and Z. They are bosons, meaning carriers of force, as is the usual photon.
At the Big Bang, the universe also had one, unified, mighty force called the Superforce ruling it. But a tiny fraction of a second before the era I am talking about, the Superforce began to break down, successively “shedding off” part of itself to make the force of gravity, and another part of itself to make the strong nuclear force, which later would be active inside the nuclei of all matter, holding quarks inside protons and neutrons once these composite particles came into being. The two forces, gravity and the strong force—important as they are—do not enter our main story today.
The remnant we have of the Superforce at the time we are talking about, a tiny fraction of a second after the Big Bang, has three forces of nature held together inside it: electricity, magnetism, and something called the weak nuclear force, which later would be responsible for beta decay, a form of radioactivity. You may remember from a physics course that “electromagnetism” unifies electricity and magnetism, as Maxwell taught us over a century ago. But, during the era I am talking about, there are really three linked forces: electro-magnetic-weak; all three are held together as the electroweak force that remained from the Superforce after it had shed off gravity and the strong force.*
Amir D. Aczel has been closely associated with CERN and particle physics for a number of years and often consults on statistical issues relating to physics. He is also the author of 18 popular books on mathematics and science.
By now you’ve heard the news-non-news about the Higgs: there are hints of a Higgs—even “strong hints”—but no cigar (and no Nobel Prizes) yet. So what is the story about the missing particle that everyone is so anxiously waiting for?
Back in the summer, there was a particle physics conference in Mumbai, India, in which results of the search for the Higgs in the high-energy part of the spectrum, from 145 GeV (giga electron volts) to 466 GeV, were reported and nothing was found. At the low end of the energy spectrum, at around 120 GeV (a region of energy that attracted less attention because it had been well within the reach of Fermilab’s now-defunct Tevatron accelerator) there was a slight “bump” in the data, barely breaching the two-sigma (two standard deviations) bounds—which is something that happens by chance alone about once in twenty times (two-sigma bounds go with 95% probability, hence a one-in-twenty event is allowable as a fluke in the data). But since the summer, data has doubled: twice as many collision events had been recorded as had been by the time the Mumbai conference had taken place. And, lo and behold: the bump still remained!
This gave the CERN physicists the idea that perhaps that original bump was not a one-in-twenty fluke that happens by chance after all, but perhaps something far more significant. Two additional factors came into play as well: the new anomaly in the data at roughly 120 GeV was found by both competing groups at CERN: the CMS detector, and the ATLAS detector; and—equally important—when the range of energy is pre-specified, the statistical significance of the finding suddenly jumps from two-sigma to three-and-a-half-sigma!