Sunday, July 12, 2015

"...Believe the seemingly impossible — that you can win at a number guessing game with absolutely no information".

From Quanta Magazine:

Can Information Rise From Randomness?
 Greetings, Quanta readers! It’s a privilege to present this new monthly puzzle column. I look forward to sharing some intriguing puzzles with you.

Solving puzzles is a quintessentially human activity — we are, after all, the only species that solves puzzles for fun. Like all puzzle columns, this one will primarily be an expression of that fun — a celebration of sudden insights and counterintuitive results, and an exploration of the new challenges that follow from those insights and deepen them. But in addition to fun, I also have a more serious purpose in exploring puzzles. I believe that puzzles are far more than a recreational activity. Some puzzles and their solutions can be seen as a microcosm of scientific progress, as readers build on each other’s comments and insights to discover something new, a process I experienced firsthand in many of my New York Times puzzle blogs. Puzzles can also deepen our understanding of scientific concepts, as in this puzzle whose solution provides insight into the second law of thermodynamics. And, as I argue in a recent Math Horizons article, I believe that our human love of solving puzzles is a reflection of the unique cognitive-emotional link that makes us the intelligent creatures we are. To coin a classically mixed neologism, we are Homo enigmatus — in more ways than one. Evolution has made us cognitive beings by giving us small internal rewards whenever we solve a puzzle — a very effective strategy.

So, as we solve these monthly puzzles, we’ll try to look deeper into their analytic content and their emotional denouements, but we will also search for connections between our puzzles and problems in various fields of science. We’ll also explore some fascinating overarching themes that hold inexhaustible conceptual riches. Today’s puzzle highlights two such themes: randomness and information.
Ready for our first conundrum?

Through a Random Looking-Glass
Alice laughed. “There’s no use trying,” she said: “one can’t believe impossible things.” “I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”
—Lewis Carroll, Through the Looking-Glass
Today’s puzzle asks you to believe something that seems impossible — that you can somehow win at a number guessing game in which you have absolutely no information:
I write down two different numbers that are completely unknown to you, and hold one in my left hand and one in my right. You have absolutely no idea how I generated these two numbers. Which is larger? You can point to one of my hands, and I will show you the number in it. Then you can decide to either select the number you have seen or switch to the number you have not seen, held in the other hand, as your final choice. Is there a strategy that will give you a greater than 50 percent chance of choosing the larger number, no matter which two numbers I write down?
Since you know nothing whatsoever about the two numbers, the only thing you can do, it seems, is make random guesses. But can randomness generate information?

Actually, randomness can be very powerful. When coupled with nonrandom selection and given enough time and iterations (such as in the process of evolution), randomness can exhaust almost all possible variations on a design, inevitably generating exquisite products that seem to have been planned with obsessive attention to detail. In some especially difficult problems, procedures called genetic algorithms, which simulate the interplay between randomness and selection characteristic of the evolutionary process, often provide better solutions than the best human experts. The crucial information that selection provides in this process is simple: It tells us which of two alternatives is better. Randomness, meanwhile, has the tedious job of working through a tremendous number of cases. So while randomness and selection are equal partners, the key information is provided by selection.

With no way to know which alternative is better, what can random guesses achieve? Well, you could guess correctly and, improbably, hit a jackpot. This happens to all of us from time to time, convincing some people that they can access information in unexplained ways. But such luck tends to even out in the long run. After thousands of trials, the expected outcome of guessing in our game above is exactly 50 percent, no matter how many lucky guesses you have. This seems like a dead end.

We know that randomness by itself is completely devoid of information. In Claude Shannon’s famous measure of the entropy of information, a random guess is considered to have maximum disorder and consequently an information content of zero. We decry as pseudoscience any attempt to get meaningful information out of events that seem to be random, as in astrology, palmistry, synchronicity, apparent telepathy, or Tarot card reading. The only way these methods could work is if the events are not really random but arranged in some way, perhaps supernaturally, to correspond to the information we want. And we have no reliable evidence of that in any of these cases....MORE
Also at Quanta:
New Letters Added to the Genetic Alphabet