In episode 13 of “The Lobby,” I interviewed Chris Marr, of The Content Marketing Academy. Recently, I caught an episode of his podcast. He was talking about the Rubik’s cube, and how he decided to give himself a challenge and learn to solve it. He asked for folk to share their own experiences with this cultural icon of extreme cleverness. Once I’d finished writing my answer out, it felt like a blog post. So, here’s a blog post.

It was my sister’s wedding, and on every table was a small, pastel shaded Rubik’s cube. In a way, I knew how to solve them, but right then, right there, I wasn’t sure whether or not I could. Still, it was an interesting extra layer of challenge. I picked up the little cube, nudged my brother, tapped one of the squares and asked, what color is that?

It took a while to figure out what color went where. My brother had never learned to solve one of these things, so it took us a while to work out a system that would allow him to tell me what I needed to know, without having to hear what every last square was. But eventually, roughly 20 minutes later, we’d solved it.

Imagine trying to solve the cube when you’ve been blindfolded, without a chance to look at it ahead of time, or at all. That’s me, only there’s no blindfold needed; I’m already blind.

I recall seeing a Rubik’s cube when I was a child. I could see better at the time, better but not well—I still used Braille and walked with a white cane. In fact, I didn’t realize how much I was using the sight I had left until it was gone. The cube is a good example. When I was younger, I could have looked at the colors; but by the time I was old enough to have a clue of how to solve it, I couldn’t see the cube at all, and some other solution was needed. There are “tactile Rubik’s cubes” on the market, but they cost more than I was willing to spend on a toy.

Then, one day, a friend who had become interested in the cube showed up at my apartment, and handed me one he’d modified. He had removed the stickers, drilled holes, and glued large metal staples in different patterns a different pattern on each side. I finally had a chance to play with the thing and see if I could figure out how to solve it, without looking it up and using someone else’s solution.

To use a cube solving as music metaphor—looking up the answer and perfecting it is like learning to play a difficult song; figuring out how to solve it on your own is like learning how to write that song.

I already knew that ways of solving the Rubik’s cube were on the internet for anyone with a browser, internet connection, and the will to use. It’s a test of memory and coordination, especially if you go for speed. However, I wanted to treat it as a puzzle, to find my own way of doing it.

It took me eight months. When I started, I had already heard that it cannot be solved one side at a time; it’s better solved one layer at a time instead. That saved me from going down that blind alley. Still, day after day, given a free moment, I was moving the cube around; observing the effects; Trying and mostly failing to find some interlocking set of moves that would lead to solving the damn thing already!

Then, one day I noticed that I had enough tricks to manage the overall trick. I could solve the first layer, no problem. The second layer just required one of the corner pieces of the first to go on a 7-move trip, landing where it started, wreaking havoc on the other layers along the way. That last layer was the real challenge. I actually had enough tricks to solve it for some time before I noticed how they could work together.

Let’s see, I can move these pieces in relation to one another, but they rotate. Um, oh, I can rotate them without moving them. What about the corners now. Hmm… I can move three of the corners without moving this one. Actually, that means I can move the corners until they’re in the right spot; I just have to start from different places. Rotating the corners? Easy! I found that trick by accident in the first week! Hey, I think I can…

AH HA!

Why would I bother to do it the hard way? Because, like everything else, problem solving is a skill that improves with practice.

In “The Lobby” I’ve been using my guests as part of some ongoing research. What is it that allows some people to set and reach their goals, while others give up so quickly. One of the skills that is important to cultivate is problem solving. Even more important, is your attitude toward problems and challenges.

As my friend who modified the cube put it:

step one: want to try!

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The amazing monkey manages to muster more memory than the miserable mind of man

For certain tasks involving short term and working memory, our nearest relatives do better than we do.

Check out this YouTube video to watch the chimps make us look like chumps.

Chimp vs human! – Working Memory test

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a fun feathered fact

Did you know that birds can be superstitious? No, really, I’m not making it up. In the episode, I said I’d provide one link, but I decided to give you two instead.

Here’s a short YouTube video about the experiment.

Pigeon Superstition Experiment

And here’s an approximately 13-minute lecture on the effect and its implications.

The Superstitious Pigeon: B. F. Skinner 1948 Keon West

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me think me smart me think smart

In this episode, we examine the evidence for increased intelligence do to the placebo effect.

This study found no placebo effect. In fact, expecting to do better caused people to do worse.

Cognition and the Placebo Effect – Dissociating Subjective Perception and Actual Performance

In this article, being told you had a good night’s sleep, regardless of how you felt about it, helped you do better with math and word play.

‘Placebo Sleep’ Can Improve Cognitive Skills

This paper treats the placebo effect in studies of intelligence as a problem, but through differing recruitment methods, it did produce one.

Placebo effects in cognitive training

And last, we have a paper that proposes one possible mechanism whereby a placebo could improve performance in tasks that deal with attention and working memory. They tried it and it worked, both for positive and negative effects, depending upon what the study participants were told.

The placebo effect on psychomotor performance and working memory

During my interview with Arthur Benjamin, he spoke briefly about the joy of discovering something in mathematics, even if one is relatively certain that the given relationship has been known for years, or even centuries. Along the same lines, I’ve been playing with right triangles of late.

They are called Pythagorean triples—three numbers, a, b, and c, where aSquared+bSquared=cSquared, and a b and c are all whole numbers. I wanted a way to generate them.

I started by noticing that the difference between two squares, such that one root is one less than the other, is always an odd number.

2*2=4
1*1=1
4-1=3
3*3=9
2*2=4
9-4=5
4*4=16
3*3=9
16-9=7

The result climbs up the odd number line, 3, 5, 7, 9, 11…

Another way to put this, is to say that the difference between consecutive whole number squares is on the line defined by 2x+1.

This can be shown algebraically:
(x+1)Squared-xSquared=2x+1
Expanding (x+1)Squared gives us (xSquared+2x+1)
So (xSquared+2x+1)-xSquared=2x+1
Add xSquared to each side and we get:
XSquared+2x+1=xSquared+2x+1
Take xSquared from each side and we get:
2x+1=2x+1
Take 2x from each side and we get
1=1
That’s about as true as numbers can get.

I’m not the first to have noticed that relation, but I did run across it independently.

Next, in order to confine the results to whole numbers, we substitute some whole number that is greater than or equal to 1 for x, which will give you side a.

2*2+1=3

Next we square a: 3*3=9
Subtract 1: 9-1=8
And divide by 2: 8/2=4
That gives us side b=4.
Next since we know the difference in length from b to c is 1, c=b+1: 4+1=5
3, 4, 5
3*3=9
4*4=16
9+16=25
5*5=25

Let’s try using 2 in place of x. 2x+1
2*2=4, 4+1=5, a=5
5*5=25, 25-1=24, 24/2=12, b=12
12+1=13 c=13
5, 12, 13
5*5=25
12*12=144
25+144=169
13*13=169

This is much easier to do in your head than it is to write it out.
3*2=6+1=7
7*7=49-1=48/2=24
24+1=25
7, 24, 25
4*2=8+1=9
9*9=81-1=80/2=40
40+1=41
9, 40, 41

When I looked it up, it turned out that Euclid used this trick to prove that the set of Pythagorean triples is infinite, since the set of odd numbers is infinite and you can use any odd number to generate sides a b and c.
Q.E.D.

Next, I got to wondering if you could generalize the method, so that you could generate triples when the difference between side b and c, c-b could equal any whole number.

Yes you can! If d=c-b than the line that your numbers need to sit on is (2*d)x+dSquared, where x is any whole number greater or equal to 0, unless d=1.

If d=1, the line is 2x+1. If x=0, you end up with 1, 0, 1. And while 1*1+0*0 does equal 1*1, that’s not really a triangle; it’s a line segment.

By the by, when you use d=1 you get 1*2=2 and 1*1=1, so the line is 2x+1, which we already knew.

Let’s try d=2.
2*2=4 and 2*2=4, so our line is given by 4x+4
Put 0 in place of x.
4*0=0+4=4, so a=4
4*4=16-4=12/4=3 so b=3
3+2=5 so c=5
4, 3, 5, which is our old friend 3, 4, 5

Using 4x+4 and replacing x with 1
4*1=4+4=8, a=8
8*8=64-4=60/4=15, so b=15
15+2=17
8, 15, 17

Using x=2, 2*4=8+4=12, a=12
12*12=144-4=140/4=35, b=35
35+2=37
12, 35, 37

Now let’s set d to 3
3*2=6 and 3*3=9, so the line is 6x+9
Starting with 0
6*0=0+9=9, a=9
9*9=81-9=72/6=12, b=12
12+3=15
9, 12, 15

X=1
6*1=6+9=15 a=15
15*15=225-9=216/6=36, b=36
36+3=39
15, 36, 39

X=2
6*2=12+9=21, a=21
21*21=441-9=432/6=72, b=72
72+3=75
21, 72, 75

In fact, there are simpler methods for making triples, but I found this all on my own, and I can’t even describe the joy of it, when the numbers click together and it all makes sense.

As any whole number greater than 0 can be substituted for d, and the set of whole numbers greater than 0 is infinite, this adds another layer of infinity on top of the one Euclid found, just in case one infinity wasn’t enough
Q.E.D.

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It doesn’t have to be true to be useful

Today we talk about the placebo effect—when an inactive substance can mimic the action of medicine or other therapies.

Here’s a review of many studies and experiments done on the placebo effect.

How Placebos Change the Patient’s Brain

And here’s a link to episode 25, which I reference toward the end of today’s episode.

Ep 25: It isn’t all about positive thinking

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Lost in the mountains, with no magic

There doesn’t seem to be any one thing that could transform you into a genius, and the search is complicated by exaggeration and fabrication within the research.

Here’s a paper on where your idea of your self, and where the circuits that let you regulate said self live.

Neuroscience of Self and Self-Regulation

Here’s an article on neural plasticity, critical periods of development, and increasing the ability of the adult brain to learn.

Re-opening Windows: Manipulating Critical Periods for Brain Development

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It isn’t all about positive thinking

I had a theory, but even my personal experience suggests that there is more going on.