A hundred and one years ago, in 1913, the famous British mathematician G. H. Hardy received a letter out of the blue. The Indian (British colonial) stamps and curious handwriting caught his attention, and when he opened it, he was flabbergasted. Its pages were crammed with equations – many of which he had never seen before. There were many kinds of formulas there, and those that first caught his attention had to do with algebraic numbers. Hardy was the leading number theorist in the world – how could he not recognize the identities relating to such numbers, scribbled on the rough paper? Were these new derivations, or were they just nonsensical math scrawls? Later, Hardy would say this about the formulas: “They defeated me completely. I had never seen anything in the least like it before!”
Now, for the first time, mathematicians have identified the mathematics behind these breakthrough scrawls – shedding further light on the genius who made them.
Neuroskeptic is a neuroscientist who takes a skeptical look at his own field and beyond at the Neuroskeptic blog.
Fraud is one of the most serious concerns in science today. Every case of fraud undermines confidence amongst researchers and the public, threatens the careers of collaborators and students of the fraudster (who are usually entirely innocent), and can represent millions of dollars in wasted funds. And although it remains rare, there is concern that the problem may be getting worse.
But now some scientists are fighting back against fraud—using the methods of science itself. The basic idea is very simple. Real data collected by scientists in experiments and observations is noisy; there’s always random variation and measurement error, whether what’s being measured is the response of a cell to a particular gene, or the death rate in cancer patients on a new drug.
When fraudsters decide to make up data, or to modify real data in a fraudulent way, they often create data which is just “too good”—with less variation than would be seen in reality. Using statistical methods, a number of researchers have successfully caught data fabrication by detecting data which is less random than real results.
Most recently, Uri Simonsohn applied this approach to his own field, social psychology. He has two “hits” to his name, and more may be on the way.
Simonsohn used a number of statistical methods but in essence they were all based on spotting too-good-to-be-true data. In the case of the Belgian marketing psychologist Dirk Smeesters, Simonsohn noticed that the results of one experiment conducted by Smeesters were suspiciously “good”: They matched with his predictions almost perfectly.
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.