Life is unpredictable, and random things happen to us all the time. You might say the universe itself is random. Yet somehow, large numbers of random events can generate large-scale patterns that science can predict accurately. Heat diffusion and Brownian motion are just two examples.
Recently, randomness has even made the news: Apparently there’s hidden order in random surfaces, and we may be close to seeing a quantum computer generate ultimate randomness. This latter quest for perfect randomness is important because randomness brings unpredictability, and all non-quantum attempts to achieve it have the hidden flaw of being generated by algorithmic methods which can, theoretically, be deciphered. In this Insights column, we will explore how we can create randomness and defeat it in everyday activities, before soaring to philosophical heights in debating what randomness really is.
Puzzle 1: Random Combinations
Consider a simple combination bike lock like the one shown in the picture. It has three rotating discs, each of which has 10 digits embedded in numerical order. When the three discs are rotated to produce the set combination — 924 — the lock opens. When you want to lock it again, you have to scramble the digits so that they are far away from the set combination. But what does far away mean in this context? If you move each disc the maximum amount, which is five positions away, you get the number 479. But it would be easy for a tinkerer to find this position by accident, simply by moving all five discs in sync and seeing if the lock opened. Imagine that the tinkerer has enough time to try five possible combinations. In each instance, our potential thief tries out the lock after doing each of the following operations:
- Turning a single disc a random amount.
- Grabbing hold of any two discs and turning them some random amount in sync.
- Grabbing hold of all three discs and turning them some random amount in sync.
- Turning two discs different amounts.
- 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?
Puzzle 2: From Randomness to Order in Puzzles
I have often been struck by how similar solving any puzzle is to the process of science. We progress from randomness to order by adding pieces, and our confidence in the rightness of our solution is bolstered by every new piece that fits. In this second problem we will try to create a way to measure our progress as we go from a random disorderly state to a finished, orderly solution.
Consider solving a puzzle on a hexagonal grid — a honeycomb, similar to the graphene we played with in our last column. The picture on the puzzle consists of a twisting vine. Since the pattern is repeating and self-similar, you cannot be absolutely sure that two neighboring pieces belong together even though they seem to fit visually. In fact, let’s say that for each edge of a given piece, there are three possible pieces that could fit with it. So when two pieces fit together, your confidence that their arrangement is correct can only be 33.33%. However, if you can find another piece that fits with both of your connected pieces, sharing an edge with each, your confidence that this arrangement is correct will be reinforced. Let’s try to quantify how much.
- 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?
- 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?
As your clump of pieces gets larger, your confidence in it should become more unshakable. It is reasonable to assume that three isolated clumps comprising a total of seven joined pieces are no match, in terms of the confidence they inspire, to a single surrounded hexagon as described above.
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.
Puzzle 3: Is Perfect Randomness Possible?
For part three of this puzzle, I give you the randomness version of the famous Bohr-Einstein debates. Everyone is welcome to join in. You can join either team E (Einstein) or team B (Bohr).
In the macro world, both teams agree that mechanisms that generate randomness are only possible because of our ignorance of the forces or algorithms that drive them. 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. We have been trained to believe, according to the prevailing team B view, that this is not true in the quantum world — quantum probabilities are supposed to be objective. But is such a thing even possible? Could there not be some mechanism somewhere in the ensembles of the subquantal Planckian world that decides which of two equally likely outcomes will take place, even though we may forever lack the ability to explore that level? Even if Einstein’s nightmare vision of a deity playing dice is true, there must be an algorithm in the deity’s brain that decides every choice, no matter how whimsical or devoid of apparent reason it seems to be. Again, the randomness is due to our ignorance. It is only practically unknown, not objectively random.
The standard reply to this by team B is to say that the quantum world is just too weird to apply the rules we have inferred from our experience of the macro world. But something can be weird in two ways. It could have physical impossibilities such as faster-than-light travel, for example. This kind of weirdness could exist, it just means that we need to revise our understanding of physical law in specific circumstances, just as Einstein revised Newton’s law of addition of velocities, which becomes inaccurate at very high values.
On the other hand, something could be weird because it has logical impossibilities, such as 2 + 2 equaling 5. Such a result is impossible in any conceivable universe. Team E would argue that perfect randomness and objective probabilities are logical impossibilities. We should not accept them, but instead try to find physical mechanisms that can explain the observed results, no matter what current physical laws they break.
Where do you stand on this, Quanta readers? Are you on team E or team B? Happy puzzling — and debating!
Editor’s note: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a Quanta Magazine T-shirt or one of the two new Quanta books, Alice and Bob Meet the Wall of Fire or The Prime Number Conspiracy (winner’s choice). And if you’d like to suggest a favorite puzzle for a future Insights column, submit it as a comment below, clearly marked “NEW PUZZLE SUGGESTION.” (It will not appear online, so solutions to the puzzle above should be submitted separately.)
Note that we may hold comments for the first day or two to allow for independent contributions by readers.
Update: The solution has been published here.