# How to describe an IQ test mathematically?

Given an IQ test which consists of 3 inputs $$I_n$$ and 3 outputs $$O_n$$. From this information you are given a new input $$I_4$$ and you have to deduce the output $$O_4$$.

In other words you have to find a function $$f$$ such that $$\forall n: O_n = f(I_n)$$ and the solution is $$O_4=f(I_4)$$

The inputs and outputs may consist of sets of shapes with certain properties and relations. i.e. shapes which themselves belong to sets. e.g. "set of red things." And in turn these properties e.g. "red" belongs to the "set of colours".

The function $$f$$ changes the properties of the shapes, addds new shapes or deletes shapes.

Is there a mathematical way (using set theory) to describe this problem in terms of set theory?

A typical function $$f$$ might correspond to "make all the yellow shapes red" or "make all the red square that's above a green square orange" or "put a green square inside all the yellow circles".

But really $$f$$ is just a function going from one set of things to another.

So if we have all the information about the sets $$I_n$$ and $$O_n$$ we should be able to deduce candiates for $$f$$. (In fact we want some kind of simplest transform.)

One problem I think is that shapes could potentially belong to an infinite number of sets. e.g. we could construct new sets like "the set of things that are next to a red thing an below a blue thing".

• en.wikipedia.org/wiki/Kolmogorov_complexity – DanielV May 29 '20 at 22:18
• @DanielV Interesting. So an IQ test is really a statement of find a transformation with the lowest Kolmogorov Complexity. – zooby May 31 '20 at 2:45
• An IQ question is really nonsense, nothing more than "do you see things the same way as the person who wrote the question." But kolmogorov complexity is one way of looking at pattern matching. Note that it is very ambiguous because you can choose any computing model, especially op codes, and because if the "halting" problem is undecidable in that computing model then for some things the kolmogorov complexity will be undefined. – DanielV May 31 '20 at 2:49
• @DanielV Yes, that's true. You'd have to assume a certain finite set of 'ways of seeing thing' common to all participants. And maybe that's impossible to do. But if it is possible to do, maybe there is an algorithmic way to solve it? It might be an impossible task. – zooby May 31 '20 at 2:54
• You could always just enumerate every program, or in a non total computing environment, do the $\mathbb N^2 \to \mathbb N$ trick to enumerate all programs and steps, and check if any terminate with the desired pattern. – DanielV May 31 '20 at 2:59

I'm not sure if you'll ever have an exact model that helps with a solution. There's a lot wrong with those test questions. Relevant here: There are a lot of such riddles that are ill-defined, even if they are included in IQ tests.

Some puzzles ask you to sort the "odd one out": Here's a typical example in that it's riddl-iculously ambigous.

They presume

• a rule R: {specimens} → {0, 1}
• and exactly one odd specimen x with R(x)=0.

Since they don't give R, x, there's always ambiguity. In the linked example, each specimens consists of ~4 visual elements (horizontal lines, and left/center/right vertical lines) with values

$$A = (2, 4, 2, 3)\\ B = (3, 3, 1, 2)\\ C = (3, 4, 1, 4)\\ D = (2, 2, 0, 5)\\ E = (2, 3, 2, 5)$$

The solution they give is the rule $$5x_1 + x_3 = x_2 + x_4$$, but there is no plausible argument that this rule is special in the large set of plausible rules. Talk about degrees of freedom!

Even applying non-mathematical restrictions of plausibility, usually some real ambiguity remains.

One type that sometimes makes more mathematical sense is the following continuation type:

1▴▴, 2▸▸▸, 3▾▾▾▾, what is the next item?

Dissecting this sequence into more or less independent sequences yields the answer, 4◂◂◂◂◂

• numerical value: 1, 2, 3 - probable continutation is 4
• number of triangles: 2, 3, 4 - probable continuation is 5
• orientation of triangles: up, right, down - probable continuation is left.

My mental mathematical model of this riddle is:

graphical representation of the i-th term = some_graphical_synthesis( f(i), g(i), h(i) )

where f, g, and h are the sequences of numerical values, numbers of triangles, and orientation of triangles.

Ambiguities can arise from the graphical representation, or from ambiguities of continuation.

◻,◫,𐌎 could be 1,3,9,...,3k rectangles, or 1,2,4,...,2k non-overlapping ones. I've seen the symbol ◫ used as "two rectangles" after ◻ "one rectangle" It's not easy https://oeis.org/search?q=1%2C2%2C4%2C8%2C16

• I thought the answer to the odd one out was D because "It has no middle vertical lines". – zooby May 31 '20 at 2:56