December 21 2013

## Some Seemingly Complex IQ Questions, Their Solutions, and Techniques

I figured out I should write an article about some compelling IQ questions modeled after items in Raven’s Advanced Progressive Matrices and how they can be solved, before writing an article on the use of a number widely known as IQ to represent an individual’s intelligence.

These are the four most interesting questions I came across while doing an IQ quiz on iqtest.dk. Among the 39 questions in the quiz, they have the least intuitive solutions. Or simply put it, they are among the very few questions which I took more than 12 seconds to solve.

### Question 26

The pictures are in chronological order. Each picture (except the first one) is the result of certain transformations that are applied to the previous picture.

The transformations going on are the synchronous clockwise rotations of the center line segment, and another two shorter line segments whose pivots are at the 4th and 6th dots respectively.

Therefore the answer is A.

One of the things that make the solution not so intuitive is that when the two shorter line segments are at the edge (e.g. in the 3rd picture) and the rotations take place, instead of spinning 90 degrees and leaving the picture, they spin 180 degrees as if they are confined in the bounded pictorial two-dimensional world.

### Question 37

The pictures are in chronological order. Each picture (except the first one) is the result of certain transformations that are applied to the previous picture.

We can see that the black squares are building up diagonally, but what most people don’t notice are that the circles are both moving one block to the right per picture (and one will move to the next row if it reaches the edge), and both will be blocked by the black squares upon entering them (that is why you don’t see the white circle in the 2nd picture), and change their colors upon leaving each square (either from white to black or black to white).

Therefore the answer is H - the first dot has moved to 3rd row from 2nd row and the second dot hidden behind the black square in the 8th picture has experienced a change in color from black to white and ended up at 3rd row, 3rd column.

### Question 36

The pictures are not in chronological order. Among them there is a pattern built on the foundation that each row and column has to obey certain principles.

The 3 principles each row and column has to abide by are

1. The three pictures in every row/column must have 3 different sets of same-color block(s) that each forms a straight line and the set of 3 same-color blocks in each picture has to be in different color. (e.g. The 3 sets of same-color block(s) in the 1st picture are 1 block in light grey, 2 blocks in grey, and 3 blocks in black differed from those sets of 3 same-color blocks in the 2nd, 3rd, 4th and 7th pictures.)

2. Every row/column must have 1 picture with its sets of same-color block(s) in vertical display (the 3 pictures with vertical display are 1st, 6th and 8th), while the others’ in horizontal display.

3. In every row the horizontally displayed pictures can only contain one block in the same color as the set of 3 same-color blocks in the vertically displayed picture, while in every column the horizontally displayed pictures can only contain two blocks in the same color as the set of 3 same-color blocks in the vertically displayed picture.

Therefore the answer is F, which has 3 sets of same-color block(s) horizontally displayed with only 1 block in light grey, 2 blocks in grey, and 3 blocks in black dissimilar from the other pictures’ in the third row and column.

### Question 39

This question can be solved by viewing either that among the pictures there is a pattern built on the foundation that each row and column has to obey certain principles, or that each picture (except the first one) is the result of certain transformations that are applied to the previous picture.

The first theory I formulated that works is that in each row/column the different sets of similarities in terms of the position of polygon those 3 pictures share with their corresponding neighboring pictures are of the same number. (e.g. The first and second pictures have 4 similarities - they both contain two triangles in the 4th and 6th blocks, a cross in the 7th block and a circle in the 9th block - and 4 is the number of similarities the second and third pictures have.)

Therefore the answer is B, the only picture that shares 1 similarity with its horizontal neighbor, the 8th picture (which also shares 1 similarity with the 7th picture), and 1 similarity with its vertical neighbor, the 6th picture (which also shares 1 similarity with the 3rd picture).

But I was not quite satisfied with this theory as it doesn’t really describe the pattern in great details and it allows more than one possible answer to exist (though among the choices the only possible answer is B). So I went on studying the pictures and managed to come up with a more elaborate theory that works. Every picture undergoes a set of transformations to turn into the next picture.

When the next pictures are in the same row, the transformations consist of

1. the transfiguration of shapes that causes triangles in the original picture to become crosses, circles to become triangles and crosses to become circles
2. the one-block-to-the-right-per-picture movement of all polygons (and one will move to the next row upon reaching the edge, or back to the first block upon reaching the end).

And when the next pictures are not in the same row (e.g. from 3rd to 4th picture), there will be a clockwise rotational transformation instead.

Interestingly, this theory too suggests that B is the answer. And this is no coincidence. Using mathematics, we can actually prove that the pattern described in the first theory is the consequence of the transformations described in the second theory. Or in the language of propositional calculus, transformations_in_2ndT ⇒ pattern_in_1stT.

### Techniques

Now some of you may be curious. How did I actually come up with these solutions? Without further ado, here is the algorithm I use:

1. Determine if it is more likely that the pictures are formed as the result of having each row and column governed by a set of principles, or that there is a set of transformations going on in the pictures.

2. If it is a principles-based pattern, use the sub-algorithm below to formulate a principles-based theory.

1. Study the similarities between the 1st, 2nd, and 3rd pictures and pick a set of similarities that are related to one another.
2. Come up with a set of principles that these 3 pictures must abide by and use them as the explanation of why the set of similarities exists.

3. If the explanation does not work, go back to step 2, this time invent a different set of principles.

• When you realize there exist no simple and elegant principles to describe the similarities, go back to step 1, this time pick a different set of similarities.

4. If the explanation works, use it to explain the similarities in other rows and columns to test its consistency.

5. If it is consistent, look at the differences between 1st, 2nd, and 3rd pictures and observe if other rows and columns exhibit similar differences.

6. If it is not consistent, go back to step 2, this time invent a different set of principles.

7. If they exhibit similar differences, come up with principle(s) that explains this set of differences and add these/this principle(s) to the set of principles that pictures in each row and column must abide by.

• If you are taking too long to devise a principles-based theory that works, go back to step 1, this time study other correlated pictures, or consider the possibility that the pattern can be described by a transformations-based theory.

3. If it is a transformations-based pattern, use the sub-algorithm below to formulate a transformations-based theory.

1. Study the differences between the 1st and 2nd picture and try to make sense out of them.

2. Come up with a set of transformations the 1st picture has to undergo in order for the differences to exist in the 2nd picture.

3. If you fail to make sense out of their differences, study their similarities and come up with a set of transformations the 1st picture has to undergo in order to keep the similarities unaltered but result in the differences in the 2nd picture.

4. If the explanation does not work, go back to step 2, this time invent a different set of transformations.

• When you have invented a number of transformations and still have not managed to successfully explain why the differences are there, go back to step 1, this time study other correlated pictures, or consider the possibility that the pattern can be described by a principles-based theory.

5. If the explanation works, apply the transformations to the 2nd picture and see if it outputs the 3rd picture. If it does, try deriving the 4th picture by applying the transformations to the 3rd picture, etc. If it doesn’t, go back to step 2, this time come up with a different set of transformations.

• If you are taking too long to devise a transformations-based theory that works, consider the possibility that the pattern can be described by a principles-based theory.

4. Go through all choices of answer and see which one fits the theory. If more than one fit, or none fits, or you are not satisfied with your theory, go back to step 1.

But most importantly, you have to be confident and always remember that as a human with a well-functioning brain, you are capable of solving logically constructed questions using logic. It is just a matter of time. And then just keep trying, old sport. It is this frame of mind that gets you a higher score in an IQ quiz. And succeed at anything in life.

(I believe Feynman would have used a similar algorithm to solve those questions. And probably a more efficient one.)

Tagged: IQ, algorithm, logic, pattern