Quanta Magazine
www.quantamagazine.org
Puzzle 1: Random Combinations 1. Turning a single disc a random amount. 2. Grabbing hold of any two discs and turning them some random amount in sync. 3. Grabbing hold of all three discs and turning them some random amount in sync. 4. Turning two discs different amounts. 5. Turning all three discs different amounts. Our puzzle question is: If the code to open the lock is 924, what set of scrambled numbers is most resistant to this random tinkering, and how many such combinations are there? What’s the probability that the code will be found? The solution below is based on a clarification I made that in actions 4 and 5 the discs are not necessarily turned the same amount. Also, we will assume that in actions 1 to 3, it is possible for the thief to turn the specified discs all the way around (a turn of 10 — or zero — digits, returning the disc or discs to the initial position). With these assumptions in place, let’s calculate the probability of each of the thief’s actions. One thing to note is that any action that the thief may perform randomly on a disc to reach a particular number is potentially reversible by performing a complementary, equiprobable action on the same disc. Thus, there is a probability of 1 in 10 that moving the leftmost disc randomly will get us from 924 to 624, and there is also a 1 in 10 probability that moving the same disc randomly will get us back from 624 to 924. This is true whether we move one, two or all three discs randomly. So in order to determine how many scrambled positions will allow the thief to solve the lock by doing a particular action, we can conveniently start from the set position (924) and count how many three-digit combinations can be reached with that action. Starting from 924 and turning a single disc, you can reach three-digit combinations of the form x24, 9x4 and 92x, where x represents any of the 10 digits. There are 10 of each of these. However, the original combination 924 is redundantly included in the last two forms as well, so there are a total of 10 + 9 + 9 = 28 such combinations. If the lock is scrambled to any of these combinations, the thief has a 1/28 chance of opening the lock with this action. Turning two discs in sync makes possible combinations of the form 9##, #2# and ##4, where the two # signs represent the amount (the same for both) that those two discs are turned from their original setting. As above, there are 28 such combinations, again giving a 1/28 chance of success. Turning all three discs in sync allows access to the 10 combinations resulting from global turns of 1 to 10 digits — 035,146, 257, 368, 479, 580, 691, 702, 813 and 924 — giving a 1/10 chance of success. Turning two discs randomly, but not necessarily the same amount, gives access to all combinations with a 9 as the first digit (numbers 900 or greater), all combinations with a 2 in the middle (x20s), and all combinations that end in 4. There are 100 of each of these. However, 10 of the x20s were already counted with the 900s, and there are also 10 combinations ending in 4 in the 900s and nine others among the x20s. So the total number of combinations is 300 − 10 − 19 = 271 for this step, with a 1/271 chance of success. Turning all three discs at random allows access to any three-digit combinations, each time with a 1/1,000 chance of success. Puzzle 2: From Randomness to Order in Puzzles 1. You find three pieces that appear to fit together without obvious misalignment in the vine pattern across the shared edges. What is the measure of your confidence that this arrangement is correct? 2. You find a central hexagonal piece surrounded by six others, and they all seem to fit with one another. What’s the measure of your confidence in the correctness of this pattern? The third part of this puzzle question is open ended and is an attempt to quantify the above difference. Can you come up with a measure of the degree of completeness of a partially solved puzzle? Your method should be able to assign a number between 0 and 100 to any partially completed 10-by-10 hexagonal puzzle. The number should represent a degree of completeness that roughly correlates with the proportion of the final solution that you would expect to be correctly represented thus far. Abhinav Deshpande answered the first two questions as follows: For three tiles in a triangle-like orientation, we have p = (2/3)3 since there are three edges that can be deleted, and each edge is deleted with probability 2/3. This gives 1 − p = 0.7037, i.e. 70.37% confidence. For a hexagonal arrangement, there are 6 + 6 = 12 edges that can fail, giving 1 − p = 1 − (2/3)12 = 0.9923 or 99.23% confidence. Deshpande took this idea to another level, proposing a measure using logarithms and connecting it to the concept of entropy, which, as he states, is a natural measure of disorder and randomness. His measure for a 10 × 10 grid is n − 100 × (log m)/(log 100), where m is the number of alternative layouts, and n is the total number of tiles already placed. You can read his description of the various advantages of this measure here. Puzzle 3: Is Perfect Randomness Possible? RRG nailed my motivation for proposing this debate, writing: In quantum mechanics, if we consider the standard double slit experiment, we can’t predict where any individual particle will hit the screen, but we can predict the probability of where it will hit the screen. And these probabilities can be extremely accurate and reliable. This reliability and accuracy of probabilities is significant evidence that something is causing it. It is analogous to thermodynamics. We can measure the temperature in a room very accurately without knowing what each molecule of air is doing. Similar to the probabilities in quantum mechanics, the temperature emerges from a deeper level of physics. My thoughts exactly! Why does a particular particle passing through a double slit impinge on the detector at, say, the upper left of the screen, instead of the lower right? Some causal chain of events (perhaps subspace quantum gravity mass-energy fluctuations) must have caused this particular choice of location in this particular instance. If that’s the case, quantum randomness is not a perfect, objective and magical part of the universe, but a consequence of our ignorance of the underlying physics, just as classical randomness is. Granted, as Mark Thomas observed, the probability space determined by Planck mass-energy may be huge. It might be large enough to achieve close to perfect randomness in the mathematical Kolmogorov sense (thank you, Ruslan Ciurca, for the link to your description of Kolmogorov complexity and randomness). But if that’s the case, the Schrödinger equation is simply an approximate, ensemble equation and should not be treated as sacrosanct, nor should it form the basis for the currently popular “many worlds” interpretation based solely on considerations of mathematical simplicity. This latter approach has been championed by the physicist Sean Carroll. Rob McEachern took exception to my statement: “If you knew all the forces that were acting on a flipped coin or a rolled die, you could, with the requisite computing power, predict what the final outcome would be.” McEachern wrote: That statement is false. You must also know all the relevant initial conditions. And therein lies the problem. In any complex situation, the information content of the initial conditions, is vastly greater than the information content of the forces, or laws of nature. Consequently, it is vastly more difficult (often impossible, even in principle) to acquire the necessary information regarding the initial conditions, than it is to acquire perfect knowledge of the laws. I agree that it is impossible to acquire perfect knowledge of initial conditions to infinite decimal places. But I think most physicists would agree that it is possible to achieve the relevant knowledge with the accuracy required for something like a coin flipped indoors, and therefore to predict its outcome in most instances. Of course, this may not be possible if a hurricane-force wind burst into the room and literally initiated chaos. Now, it may be that the aforementioned Planck mass-energy fluctuations are indeed like hurricanes that constantly sow chaos and that this is the true cause of quantum randomness. But even then, in principle, there is a deterministic causal chain. Team E would argue that we’re just ignorant about the details. As with the other two questions, Abhinav Deshpande gave a superb, balanced, exhaustive and well-reasoned description of the current state of the art in this field, with references to some very good articles. Deshpande correctly states, “I don’t think the founder of relativity was comfortable with [nonlocality] either (even if the nonlocality doesn’t actually permit information transfer faster than the speed of light).” That’s true, but remember that Bell’s theorem was proved a decade after Einstein passed away. Faced with the strong experimental confirmation of the Bell inequality, the modern Team E has no choice but to modify Einstein’s original views and accept that nonlocality is a fact and that “spooky action at a distance” does happen. That means it’s possible that there are supraluminal or extra-spatial connections among the components of an entangled quantum object even though external information transfer is limited by the speed of light as relativity dictates, and the nonlocality never leaks out into the open. That’s all for now, but we are not done with randomness yet. This month’s prize goes to Abhinav Deshpande. Many thanks to all who contributed, and see you next month for new Insights. Subscribe