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Comment by tlb | Hacker News Reader

An interesting category of problems are like Bongard problems in that you have to deduce the rule from examples, but the examples are presented one at a time at random long intervals so you have to work from memory. Most real-world learning is like this.

When working from memory, it's normal for your memory to have already parsed the previous situation into features. As some of the later examples in the blog illustrate, it's easy to fall into parsing examples into the wrong set of features, which is how you'll remember them.

While I could solve all the problems in the article, I doubt I could solve any but the simplest if I was shown 1 image per day over 12 days and not allowed to write anything down.

Perhaps the lesson is that when you're trying to deduce a rule (say, for what conditions your software crashes in) you can increase your rule-discovering power greatly by making notes and being able to look at several examples side-by-side.

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This is exactly right, and it is exactly what makes quantum mechanics and relativity so hard to wrap your brain around because by the time you get around to learning them you have almost certainly deeply internalized a classical model of the world. It's just obvious that classical mechanics is "correct", that the world consists of objects embedded in a three-dimensional space that exist in specific places at specific times, and even talking about a world where this is not true doesn't even make sense, let alone qualify as a viable candidate for actual truth.

It is equally "obvious" that the heavens are governed by different laws of physics than the earth, because things on earth fall down if unsupported and naturally come to rest and things in the heavens don't. And of course all of these things are equally wrong.

One can and should apply the same lesson to social and political statements. For example, people get hung up on arguing about things like whether or not "God exists" as if they were arguing about a question of objective fact when actually what they are arguing about is the meaning of the words "God" and "exists."

I wrote a longer take on all this about six years ago:

http://blog.rongarret.info/2015/02/31-flavors-of-ontology.ht...

You also have to know you want to solve a problem.

Once you get to the point where you have any hypothesis whatsoever, no matter how weak, a systematic approach (saving examples as test cases) helps to avoid confirmation bias and makes testing further hypothesis less costly.

Another hard one is when there is a simple, probabilistic rule. You usually end up with an over-complicated rule to cover all your data instead of the true rule. (Of course that gets down to what is at the basis of the probability: are you satisfied with a probability?)

Probabilistic rules themselves tend to require much more data, which can be expensive.

In computing we try to write deterministic tests that either pass or fail, which means you can run them once after a change and know what the state is. Even if you just suspect flakiness you may have to run the test hundreds of times to be confident that the probability of failure is sufficiently low.

Here's a deduce-the-rule problem that completely stumped me until I wrote down a series of examples: https://illuminations.nctm.org/lessons/petals/petals.htm

It seems trivial from the name and I always get the same answer as it gives; is there a way to confirm my rule?

Spoiler?, ROT-13: Zl vagrecergngvba bs gur ehyr: "Crgnyf nebhaq gur ebfr" zrnaf pbeare qbgf nebhaq n pragre qbg, fb gjb sbe ebyyvat n guerr naq sbhe sbe ebyyvat n svir; mreb sbe nyy bgure ebyyf, nf gurl unir ab pragre qbg. Lrf, gur vafgehpgvba nobhg gur anzr orvat vzcbegnag jnf fb üore-boivbhf gung guvf jnf zl ulcbgurfvf rira orsber zl svefg tynapr ng gur qvpr.

> While I could solve all the problems in the article, I doubt I could solve any but the simplest if I was shown 1 image per day over 12 days and not allowed to write anything down.

This is true for me too. Even 1 image every minute over 12 minutes would be quite difficult for me. I'm not sure I could solve any of these problems without being able to look at the figures side-by-side, back-and-forth, again and again, until I get an "aha!" moment.

On a related note, I have found that revisiting old information multiple times over sometimes longish time spans and unconsciously comparing it with newer, fresher information often helps me gain insight and greater depth of knowledge.

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