The Haselbauer-Dickheiser Test can be found at http://matrix67.com/iqtest/.
In this blog post, I will study Question 8 from this test.
The question is about discovering a pattern among line segments. There are many crossings among these line segments and each crossing is made by only two line segments.
Below I present the original question for your convenience:
Please do not read the rest of this article, if you want to attempt to solve this question on your own. The rest of this article describes my attempt at solving this question and you should not read it, unless you want to or you do not mind coming across relevant ideas, spoilers, hints, solutions, and strong opinions concerning this test.
You have been warned and I now consider that you continue to read knowing that what you come across for the rest of this article may forever spoil things for you and/or present strong opinions against this test.
Last warning: please do not read this blog post, unless you are certain that you know what you are doing. If you are not sure, then it would be best if you stopped reading at this point.
OK. If you are here, it means that you want to know my opinion. Well, ok then!
To cut a long story short, my opinion is that the test is highly inappropriate. In this blog post, I will focus on the study of question 8.
I do not know the answer to this question, but I promise you, dear reader, that if you continue reading the rest of this blog post, I will make every effort to make it useful to you.
First of all, I will discuss about the question. The question wants us to find the pattern behind the letter sequence that stems from the image.
It is the same as the person who posed the question is asking us to find what she is thinking. “What am I thinking?” This question may come from a child, since a child is immature. It may also come from a mentally retarded person. It may also come from a person who has the incorrect impression that intelligence can be measured and that people that happened to be good at guessing are intelligent. It may also come from sociopaths. Knowing and not revealing may be a good indication of a sociopath.
How can we handle such a question? First of all, we should not be handling it. It is not a proper question. It is not a valid question. It is a question a child would ask because of her immaturity. I always say to my students that there are no stupid questions, that any honest question is valid, has merit and deserves to be answered honestly. I say this to my students because no student ever asked me: “What am I thinking?” If a student ever asked me this question, I would certainly tell her that she is being silly.
This question wants us to find the reason behind the letter sequence and the reason stems from the image. The image depicts line segments. Not lines, but line segments. They each have a name, each is named by a different letter of the alphabet from A to Q. And there are a lot of crossings, each crossing by two of the segments. No three or more segments cross at a point. The closed areas that are formed are painted and I think that this is very important. This is done in order to exclude line segment L from the sequence.
Let me explain this important point.
As you can see, line segment L crosses line segment J, but no closed area is defined there, thus no color is applied. This is the clue that denotes that we should not take this line segment under consideration and that our attention should instead focus on the colored areas.
But I am not expressing it correctly. What I am really trying to say is this: The pattern that is given to us is Q P O N M. So, these are the letters backwards. So we would continue the sequence as follows: Q P O N M L K J. And this would be correct, because we would think that all line segments participate in the image, and the simplest explanation is that we take the letters in reverse order.
Occam’s razor. We will accept the simplest explanation. But here, line segment L is obviously the odd one out. It creates a crossing that does not correspond to a painted area. And this is shown in a very conspicuous way. Thus, the creator of the question definitely wants us to steer away from choosing L as the next line segment.
It is important to point out that the simplest explanation that takes all evidence into account is the best explanation. Thus, if we find two models that explain all evidence, then we should choose the simplest one. But of course, we must not discard any evidence. So, the alphabet letters in reverse order is too simple a model and does not encompass all evidence from the image. So, we should discard it.
Let us see if we can find other models that account for all evidence. If we can, then the simplest ones should be candidates for the solution.
Oh, and I have to stress another important point. The pattern we find has to produce a sequence starting with Q P O N M and it has to do it in a unique manner for any member of the sequence. It has to be unambiguous what each member in the sequence should be, starting from Q and going all the way to the last member of the sequence.
Let me begin by counting the number of crossings that each line segment makes.
Below I present the same sequence, sorted from the most crossings to the least crossings and with a secondary sort of reverse alphabetic order.
If we take the even numbers from the above sequence, we have the following sequence: Q P O N M J D K.
Let us see if we can find any other pattern that produces a sequence beginning with Q P O N M and unambiguously defines each member of the sequence up to the last.
We start from the last alphabet letter that exists: Q. What line segments does Q cross? We can find them easily and we can order the alphabetically and we can take the last one. It is P.
Now we take P. What line segments does P cross? We order them alphabetically and take the last one not already in the sequence. It is O.
Now we take O. What line segments does O cross? We order them alphabetically and take the last one not already in the sequence. It is N.
Now we take N. What line segments does N cross? We order them alphabetically and take the last one not already in the sequence. It is M.
Now we take M. What line segments does M cross? We order them alphabetically and take the last one not already in the sequence. It is G.
Now we take G. What line segments does G cross? We order them alphabetically and take the last one not already in the sequence. It is K.
Now we take K. What line segments does K cross? We order them alphabetically and take the last one not already in the sequence. It is J.
So the sequence produced is Q P O N M G K J.
Ok, we found two different and simple patterns that produce an unambiguous sequence. The first pattern sorts the line segments by decreasing number of crossings, then in reverse alphabetical order and discards the odd numbers. The second pattern begins frm the last letter assigned and for each line segment takes the biggest letter whose line segment it crosses. Which of these two patterns is the simplest? It is subjective. Is there a simpler pattern that requires even less creativity from our part? Hard to say.
Here is another pattern: Let us begin from Q. Again we will find the line segments that Q crosses. We will order them alphabetically and will take the last two and list them in reverse alphabetical order. So from Q we get P O.
Then we take O. We find the line segments that it crosses and are not yet part of the sequence, we order them alphabetically and take the last two.
So from O we get N M. And so on.
So, this is another pattern that produces Q P O N M unambiguously and can continue to produce unambiguous members of the sequence. And other patterns such as the above can be found. And I stress the fact that they can produce the sequence members in a n unambiguous manner. But since they are more elaborate, more creative and more complex, we should discard them.
The same goes for taking into consideration the number of areas attached to each line segment or the corners of the areas attached to each line segment. They are quite complex and do not correspond well to the sequence Q P O N M. But, if you want to try for yourself, below I present an image depicting the number of edges for each one of the areas.
Since there are already many simpler models producing the pattern Q P O N M in an unambiguous and unique manner, I believe that the image above should be of no help to you, but I provide it in case you want to study it.