The Rubik's Cube

Solving the world's greatest puzzle

The Problem

Several weeks ago, my daughter asked me to teach her how to solve a Rubik’s Cube. Despite having dabbled with the toy, as most people probably have at one point or another, I myself didn’t know the solution. I’d always been hesitant to be told or to look it up; it seemed like a problem I could solve on my own with enough focus and determination.

Yet, realistically, I’d never sit down with that requisite focus and determination. And my daughter expressing interest in it was the catalyst I needed to finally give in.

We watched a four part video series (this one, in case you’re interested). After two weeks of persistence, discussion, and practice, we were both able to consistently solve any starting configuration in about two minutes.

This image shows a configuration that would actually never be possible on a Rubik’s Cube. Do you know why?

The Solution

There are many fascinating aspects of the Cube. But I think one particular fact trumps all:

The number of possible configurations of the cube is 43,252,003,274,489,856,000.

But the Cube’s “God’s Number”, or the number of steps it takes to solve any of those 43 quintillion configurations… is only 20.

Let that sink in for a moment. Any of the 43,252,003,274,489,856,000 possible ways the colors on a Rubik’s Cube can be distributed can be solved in less than two dozen moves. This has been mathematically proven (as recently as 2010).

Once I learned this, I couldn’t help but draw several thought-provoking parallels and conclusions to other parts of life. For instance:

  • Most people attempt to solve the Cube one side at a time. Once you know how to solve it, you realize that this approach leads to more dead-ends than not. In fact, the correct way to solve the Cube is to think about it in layers (something that video series does a great job of showing). Yet, ask the average person on the street how to start solving one, and most will probably advise you to just solve one color at a time. The conclusion? Not only is a novel approach sometime necessary to solve a problem, but it is often wise to ignore the prevailing wisdom of other people. Problems should be solved logically and from first principles, despite what the masses say.

  • Consider that any problem we’re presented with (not just in the Cube, but in life) has significantly more wrong answers than right answers. This is essentially the Second Law of Thermodynamics (also called entropy), a topic that I’ve always been fascinated with and have written about before here and here. The Rubik’s Cube may be the most perfectly encapsulated representation of entropy ever created. Any action, on average, increases the randomness of the Cube and takes you farther from an ordered state. That’s exactly what makes it so challenging to solve.

  • Until I finally knew how to solve the Cube, I didn’t appreciate that an underlying algebra governs the 3×3×3 toy, and—once grasped—makes it much easier to understand. The use of the word algebra here is intentional; the Cube perfectly showcases concepts of a field of math called abstract algebra. The Cube itself is a mathematical group, which leads to some fun realizations:

    • Any sequence of moves has an inverse set of moves. You can use this to start tackling some unique challenges. Given a solved Cube, how can you then get it to a different particular configuration, such as swapping certain corners or creating checkerboard patterns?

    • Any sequence, repeated enough times, will return the Cube back to its original configuration. Lose track of what you’ve been doing? Repeat a sequence to eventually return to where you started (it just might take a very long time).

    • The group has sub-groups, meaning there are ways to break the entire Cube into component parts that are isolated. For instance, edge pieces have two sides. Corner pieces have three. Each can therefore never be substituted for the other. Realizing this makes the Cube easier to solve, because it reduces the overwhelming total number of faces to different categories. When solving corners, it’s comforting to know that there are only eight possible places each one can be. This notion of finding ways to simplify tough questions reminds me of the concept of Bounded and Unbounded Problems, which I wrote about a few weeks ago.

  • I was wrong all those years during which I refused to look up how to solve the Cube. So dead-set on figuring it out myself, I missed out on a key discovery: The beauty of the Cube is not in knowing how to solve it. It’s in knowing why it works. It’s in dissecting the steps used to get to the answer and appreciating all of those implications I’ve outlined above. The takeaway here? There’s value in humbling yourself and allowing someone smarter to teach you things you didn’t know.

Final word goes to Max Park, who only last year broke the world-record for solving a Rubik’s Cube: 3.13 seconds. Here’s the video of him doing it:

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