Solving a Rubik’s Cube tests key mind skills – memory and visual thinking and sometimes challenges in life can be overcome.
It took Ernõ Rubik more than a month to solve his namesake puzzle the first time. Today, competitive cuber’s can best the classic brain teaser in less than five seconds, and casual players can do it in minutes. Their not-so-secret weapon is math. More specifically: algorithms. Devising or memorizing sequences of moves that accomplish a particular goal—for instance, swapping two corners—is key to cracking your Rubik’s Cube. When game designers start stacking more layers onto a standard 3-by-3-by-3-square cuboid, it doesn’t change those algorithms much; it just makes the solve mega-tedious. But changing other variables like rotation angles and block depths creates puzzles for many skill levels and breaking points.
There are 43,252,003,274,489,856,000 ways to solve a Rubik’s Cube. That’s just over 43 quintillion for the less numerically minded. But don’t try to figure them all out – at a rate of one turn per second, it would take you 1.4 trillion years to make your way through all the configurations!
‘When you are studying from a book, lots of people go straight to the end to look for the answers. But that’s not my style. For me, the most enjoyable part is the puzzle, the process of solving, not the solution itself. Erno Ribik.’
What’s the answer to the ultimate question of life, the universe and everything? In Douglas Adams’ science fiction spoof ‘The Hitchhiker’s Guide to the Galaxy’, the answer was 42; the hardest part turned out to be finding the real question. I find it very appropriate that Adams joked about 42 because mathematics has played a striking role in our growing understanding of the universe.
The idea that everything is, in some sense, mathematical goes back at least to the Pythagoreans of ancient Greece and has spawned centuries of discussion among physicists and philosophers. In the 17th century, Galileo famously stated that our universe is a “grand book” written in the language of mathematics. More recently, the Nobel laureate Eugene Wigner argued in the 1960s that “the unreasonable effectiveness of mathematics in the natural sciences” demanded an explanation.
Soon, we’ll explore a really extreme explanation. However, first we need to clear up exactly what we’re trying to explain. Isn’t math all about numbers? You can probably spot a few numbers here and there but these are just symbols invented and printed by people, so they can hardly be said to reflect our universe being mathematical in any deep way.
When you look around you, do you see any geometric patterns or shapes? Here again, human-made designs like the rectangular shape of a book or a magazine don’t count. But try throwing a pebble, and watch the beautiful shape that nature makes for its trajectory.
The trajectories of anything you throw have the same shape, called an upside-down parabola. When we observe how things move around in orbits in space, we discover another recurring shape: the ellipse. Moreover, these two shapes are related: The tip of a very elongated ellipse is shaped almost exactly like a parabola. So, in fact, all of these trajectories are simply parts of ellipses.
We humans have gradually discovered many additional recurring shapes and patterns in nature, involving not only motion and gravity, but also electricity, magnetism, light, heat, chemistry, radioactivity and subatomic particles. These patterns are summarized by what we call our laws of physics. Just like the shape of an ellipse, all these laws can be described using mathematical equations.
Equations aren’t the only hints of mathematics that are built into nature: There are also numbers. As opposed to human creations like the page numbers in this magazine, I’m now talking about numbers that are basic properties of our physical reality.
For example, how many pencils can you arrange so that they’re all perpendicular (at 90 degrees) to each other? The answer is 3, by placing them along the three edges emanating from a corner of your room. Where did that number 3 come sailing in from? We call this number the dimensionality of our space, but why are there three dimensions rather than four or two or 42?
There’s something very mathematical about our universe, and the more carefully we look, the more math we seem to find. So, what do we make of all these hints of mathematics in our physical world?
Let’s look at mathematics in a wider context:
1. Science mostly frames data in mathematical relationships. But physicists like Joscha Bach are updating that nature “written in mathematics” picture, repainting the universe as “not mathematical, but computational.”
2. “Computation is different from mathematics.” Math mostly isn’t computable ( = unsolvable). But matter computes (it always knows what to do).
3. For Bach, physics is about “finding an algorithm that can reproduce” the data. He calls this computationalism, but “algomorphism” better emphasizes algorithmic structure.
4. Algorithms are detailed instructions, recipes that specify every ingredient and processing step. Beyond Bach’s desire for computability, algorithms can better express critical properties of sequence and conditionality.
5. The algebraic equation language (AEL) that physicists are trained to love has key limitations (classic case “the 3 body problem”).
6. Deeper consequences lurk in AEL’s grammar. X + Y = Y + X, but cart before horse ≠ horse before cart. Sequences often matter (in life, even if not in AEL syntax).
7. Some seek only AEL. Sabine Hossenfelder challenges anyone “to write down any equation … that allows … free will.” Perhaps AEL can’t paint the needed picture?
8. Freeman Dyson says “the reduction of other sciences to physics does not work.” Living cells aren’t best viewed just “as a collection of atoms.”
9. Your bag of atoms, to be you, takes mind-bogglingly complex processes, orchestrating trillions of ingredient atoms (= massively sequential, utterly algorithmic, not algebraic).
10. Biology also needs algorithmic logic because life unavoidably involves choosing (like choosing what to avoid to avoid being eaten). Algorithms provide a language naturally fit to describe choosing. AEL can’t easily express rules like, “If predator, then run; otherwise graze.”
11. Natural selection is itself a meta-algorithm. Likewise economics (~productivity selection) is deeply algorithmic (sadly its modelers mainly write AEL).
12. The universe abounds with algorithms in action. Physics has mostly painted AEL-suited pictures. But life expresses richer logic in its empirical patterns.
13. Choosing is key (as is choosing the right language). Even non-living systems — e.g., computers — embody choosing logic.
14. Babies, of necessity great causality detectors, distinguish two pattern types — physicsy things (=unchoosing) from what’s living (=exhibits “contingency patterns”).
15. What if systems could be described by a “choosing quotient,” CQ, that works sorta like electric charge. Things with electric charge (net charge > 0) do things that things without it don’t. Perhaps CQ > 0 systems can use energy to respond differently than physics’ CQ=0 systems?
16. Causation itself could be pictured as that which enables transitions between algorithmically computable states.
17. AEL can’t usefully paint all empirical patterns. Algorithms provide a richer palette.
The Mathematical Universe Hypothesis, which states that our external physical reality is a mathematical structure, to answer this question….we need to take a closer look at mathematics. To a modern logician, a mathematical structure is precisely this: a set of abstract entities with relations between them. This is in stark contrast to the way most of us first perceive mathematics — either as a sadistic form of punishment or as a bag of tricks for manipulating numbers.
Modern mathematics is the formal study of structures that can be defined in a purely abstract way, without any human baggage. Think of mathematical symbols as mere labels without intrinsic meaning. It doesn’t matter whether you write “two plus two equals four,” “2 + 2 = 4”.
The notation used to denote the entities and the relations is irrelevant; the only properties of integers are those embodied by the relations between them.
In summary, there are two key points to take away: The External Reality Hypothesis implies that a “theory of everything” (a complete description of our external physical reality) has no baggage, and something that has a complete baggage-free description is precisely a mathematical structure.
The bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Everything in our world is purely mathematical — including you.
As Erno Rubik once said:
“If you find a solution with the Cube, it doesn’t mean you find everything. It’s only a starting point. You can work on and find something else: you can improve your solution, you can make it shorter, you can go deeper and deeper and collect knowledge and many other things.”