Analogies.

I would argue that the ability to understand, produce and use analogies as both a tool of communication and a way to grasp and sort concepts is a defining feature of the human mind and of humanity in general.

No other organism seems to exhibit such a unique capacity to abstract away relations between different objects and reapply them in other contexts, mostly in a "satisfying" manner. This "satisfaction" alone seems to be unique to humans and highly correlated with other abstract notions like beauty.

Before getting to any actual example problems, I would like to address some general topics regarding classification of such problems, solving-approaches and other important clarifications.

No Algorithm

The main reason solving analogies poses such a unique challenge is ,at least in my view, due to not having one unified procedure that would allow categorizing every possible relation between two (or more) arbitrary objects. In other words; it is very hard to make analogy-solving mechanical as also evident by actually trying (note that there was also some partial success).

We can clarify this point even further by trying to classify the actual features that make analogies as a general domain of problems so challenging:

1. Category Boundlessness

Meaning that analogies can be formed under any category; verbal, visual, mathematical, tactile, animal-related, food-related and so on.

2. Instance\Example Free-Form

Meaning there is no "correct" example or "instance" to present a relation with. If the logical relation presented is "increment by 2" then the instance

"17:19" is just as valid and just as representative of the the rule as "Square:Hexagon". Both the relation and the instance are related to quantity, regardless if the instance is taken from a numerical category or a sub-verbal-category, and so the idea is preserved.

3. Experience\Culture Hyper-intimacy

Meaning that an analogy can (but not necessarily) heavily relay on prior very specific knowledge. Consider the following example:

(The form "A:B::C:D" means that the relation between A and B also applies for C and D)

For you, a human who grew up and still living within human culture, it might be easy to see that the answer we are looking for is the face of Leonardo Dicaprio but think for a second about a computer program trying to crack this. It would have to have:

-Access to human cinema

-Understanding of English

-Understanding of relation between title and film

-Understanding of the concept of main actor (possibly by correlation with screen time)

-Knowledge about famous actors

-Facial recognition capacity

That's pretty damn specific and I wouldn't expect even some humans to be able to solve this one.

Here's another example:

Cat:Katze::Snake:?

If you're both an English and German speaker you would probably immediately recognize the solution as "Schlange", the german word for snake, but how can a non-speaker of this languages possibly be able to solve it? That's why I labeled this point as "Hyper-intimacy", because there is a profound unbreakable relationship between the analogy and an extremely narrow domain of knowledge. One can also say that a solution to such a problem might have arbitrary necessary requirements.

4. Possible Ambiguity

Another important issue is that of possible ambiguity, especially when not provided with some choices about plausible solutions. We can talk about roughly two types of ambiguity, one that concerns more visual analogies and the other numerical ones:

-Gestalt VS Analytic

Consider the following problem:

If you would allow me to armchair a bit, I would claim that the first impression for most of you is probably that the rule presented is a clockwise rotation of the internal "I" figures, But what if you are aware of the concept of modular arithmetic? Then how can you differentiate between copying the shape (while not allowing more than 3 copies) and clockwise rotation? Or, to demonstrate it visually:

The left figure refers to the "Analytic" or procedural interpretation of the rule while the right figure refers to the "Gestalt" or visuospatial interpretation of the rule. From a pure logical perspective there is no "correct" or "better" one due to having no bound on the domain of categories from which a relation might be taken. This is why it is important to give the solver a choice between plausible solutions and to make sure to also exclude a possible Analytic/Gestalt counterpart to the intended one.

Let's have a look at a very easy problem (at least for English speakers):

Hot:Cold::Inside:?

Even without providing plausible solutions to narrow down the solution space, I bet all of you were able to figure out immediately that the intended answer was "Outside". Since all of us are humans with an intimate knowledge about the English language, we also have a grasp of antonyms. But even such simple and seemingly obvious examples can quickly get you into trouble due to ambiguity of language itself and lack of context like in this example:

Pretty:Ugly::Short:?

Pretty and Ugly are clearly antonyms of one another, but what is the antonyms of "Short"? is it "Long" or "Tall"? One can't find the "correct" answer because we lack context.

-The Problem With Number Analogies

Before I'll leave you with some analogy problems you can solve on your own, I want to show another major possible cause of ambiguity. Consider the following:

4:5::7:?

A reasonable knee-jerk reaction would be that this is an easy problem and the answer is "8" since 4+1=5 and 7+1=8 but if you remember my approach to solving number series, then you might quickly realize that "11" is also a valid solution since 2*4-3=5 and 2*7-3=11. In fact so is 14, 17, 20 and any number of the form 3*A+5 since:

3*4-7=5

3*7-7=14

4*4-11=5

4*7-11=17

5*4-15=5

5*7-15=20

A*4-(A*4-5)=5

A*7-(A*4-5)=A*3+5

Yikes! Now see how by providing plausible solutions to choose from, the ambiguity problem vanishes:

4:9::64:?

1. 130

2. 185

3. 81

4. 5

You might recognize right away that this are perfect squares and then you'll have the intended solution but notice how also other reasonable solutions are nullified by having answers that are "almost correct".

I will leave you now with a few analogies to try on your own:

A.

abc:cde::ffj:?

1.vpv

2.rrl

3.hhl

4.gnn

B.

yt:af::uuhdh:?

1.eervr

2.afaff

3.nnbjb

4.lpool

C.

1331:11::8000:?

1.1729

2.3

3.16000

4.20

D.

283:716::69224:?

1.19000

2.30775

3.44228

4.138448

E.

F.

G.

And that's all for now! I hope you enjoyed yourself, I hoped you learned something, solutions are coming next week so stay tuned and keep that cortex active!