I was researching intelligence quotient and IQ tests on Wikipedia. I stumbled upon, as one always does on Wikipedia, an interesting kind of IQ test: Raven’s Progressive Matrices.

It had a link at the bottom to an iq test: http://www.iqtest.dk/main.swf. It’s pretty interesting. I recommend you check it out. You know how I feel about IQ tests. So, I decided to figure out how it works. It’s actually pretty simple. I think that anyone smart can follow my simple tricks and figure out how to get a perfect score pretty easily and well under the time limit of 40 minutes.

It took me a little longer than twice as long as the test to figure out and document the general patterns as well as verify all of the answers.

## Raven’s Progressive Matrices

The puzzle is very simple, and does not even require much explanation. It simply shows you a 3×3 (or 2×2) matrix of black-and-white symbols. The lower right corner is not filled in, but the rest are. You are supposed to deduce the pattern and figure out what should most logically fill the lower right corner.

For example:

\ | /

{ | }

( | ?

What would go in the “?” spot? Good, a “)”. That’s a pretty simple pattern. They get much more complicated, but they still are all based on just a few basic rules. Please note that I made up all these terms. You don’t know which kind of matrix a given matrix is, but you can figure it out pretty quickly.

## Rules

### Momentum

Look at problem 2 on the iqtest.dk site. That’s momentum. If the first symbol and the next symbol look the same, except for one little thing moves or changes or adds to itself, and then it moves or changes or adds to itself by the same amount on the next symbol, then that’s momentum. Just follow that.

Example:

( (( (((

_ __ ___

{ {{ ?

The answer: {{{.

Note that this rule can become less obvious if there is what I call “carry”. That is, if the symbol itself is a little 3×3 matrix, and you “move” to the right, then some of the elements will fall out of the little matrix, so then you must “carry” them over to the next row of the litle matrix.

### Set Completion

Look at problem 8 on iqtest.dk. That is simple set completion. Think of each symbol having a number of properties: size, color, shape, etc. If you can’t sem to follow a progression like you can in Momentum, but it just looks like a bunch of random, but somewhat related things with similar properties, then the problem can be set completion.

I can best show you this in an example (use your imagination about the shapes):

^ O []

O [] ^

[] ^ ?

Answer: O

You need to complete the set of shapes on the last row. Notice that the last column also needs to complete the set of shapes (the diagonal too in this case, but that’s not always the case).

A common property of set-completion that makes this kind of problem much easier, is to look at the triangles made. Notice:

^ x x

x x ^

x ^ x

and

x x []

x [] x

[] x x

and

x O x

O x x

x x ?

.

Obviously, the ? should be an O. Set-completion is simple, if the first row has a red, white, and blue, and the second has one red, one white, and one blue, make sure the third has one of each.

This can get more complicated because you can have multiple properties, shapes and colors etc, all compounded on each other. But, if you just find the triangle, this problem is simple.

### Composition

If one symbol looks like the other two put together, then it is just composition. You just have to figure out in what way it should be put together. Maybe the rule is, always put it on the inside of the first. Maybe it’s, always put it on the outside. Whatever it is, this one is usually pretty easy. I won’t even give an example. Question 30 uses composition.

### Subtraction

Subtraction is much like composition, look for one thing looks like the other two put together, but with a twist. The subtraction could be complete, just one shape minus the other. Or it could be XOR (exclusive-or). You take two symbols, and take out the lines which are in one, but not the other.

Example:

_|_| | __|

|__| |_| |_|

__| |__ ?

Answer: | |

### Functions

If the first symbol in a row looks like the last symbol, but the middle symbol looks weird or especially if it’s a line or arrows or something simple, then that middle symbol might be a “function.” By function I mean something that geometrically transforms the first symbol into the third symbol. The function is not necessarily intuitive, but usually makes sense in terms of what the function symbol looks like.

In the same example I used above, the vertical bar “|” is a function that reflects the first symbol horizontally over itself, like a mirror.

\ | /

{ | }

( | ?

What’s interesting is that you can apply one function to another function. So, you might apply a rotation function with a flipping function, flipping the rotation function, creating a function that both rotates and flips. Pretty cool.

### Replacement

Replacement is where they trick you. The rule might be very simple, but it becomes very hard to figure out quickly, because the elements inside the symbol change for arbitrary reasons simultaneously. Question 25 is an example of movement with replacement together.

### Commonality

Finally, if all the symbols look randomly chosen with a bunch of properties and possible configurations, then start to look for commonalities. Don’t look for a 1.2.3. pattern like movement, just look for rules that each symbol has in common. For example, say that each black element should be on top of a white element in exactly one symbol in each row.

This can be difficult, but is easier if you know that it’s none of the other rules, and you are looking for a commonality, not a progression of patterns. Once you have some rules, start ruling out answers until you find a final answer. Question 26 is an example.

## Putting it all together

Skeptical that the answers are so easily based on the rules above? They really are. What makes the difference between an easy and a hard problem is that a hard problem will use multiple rules together. Fortunately, using multiple rules together usually doesn’t make it much harder to figure out as long as you systematically think through these possibilities. If you are having trouble with a problem, you should stop, take a deep breath, look back at the matrix as a whole, and then think through each of these rules and rule them out or use them as appropriate. All matrices follow one or more of these rules.

On the iqtest.dk site, the lowest score you can get is “below 79″. The highest you can get is “above 145.” The answers to all of the questions I put below, along with an explanation referencing the rules used to get the answers:

D Come on (momentum?)

F Momentum

B Momentum

G Set completion (angle of line)

A Momentum (size and column)

H Momentum

B Momentum (notice that the squares hide each other)

E Set completion

H Set completion / Momentum ?

A Subtraction

C Application of function (enlargement along axis)

F Set-completion (angle and number lines)

B Momentum

D Subtraction

H Subtraction

E Composition and set-completion with replacement

F Momentum (one example has carry)

C Subtraction

E Momentum with carry

D Set-completion (angle and number black/white)

G Momentum

A Oppositing? (a bit like subtraction but from sets of attributes of platonic ideal)

B Set-completion (1. small ball color, 2. big outside shape, 3. inside v. outside)

H Set-completion (1. flat bottom, 2. widening, 3. partially-closed top)

B Non-repetition? Set-completion? (Movement and replacement?)

A Commonality (180 deg rotational symmetry and middle pegs always covered)

H Subtraction

G Set-completion

E Function-application (and function-application on other functions!)

A Composition

D Subtraction yields line which is a function you apply which is reflection and delete line

E Movement

G Set-completion?

G Valuation (attach negative integer for ball inside circle, positive outside) then add

C Function application (with a bit of spatial reasoning)

F Set-completion (big-stack color, two-stack color, bar-chart position)

H Movement and replacement based on progression (replace as hidden by dark square)

F Function application (functions on functions)

B Movement with carry and replacement

## So

I think I will write a program that creates Raven Progressive Matrices dynamically. I think that would be really cool, and probably useful for some psychologists.

I just discovered that Dr. Raven has a website with links to papers describing the inner workings of the Matrices: http://www.johnraven.co.uk/pubs/pubs.html . I guess that might have made my job a little easier.