The Haselbauer-Dickheiser Test can be found at http://matrix67.com/iqtest/.
In this blog post, I will study Question 2 from this test.
The question is about giving numerical values to symbols, so that the given equations hold.
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 2.
This question is one of the most inappropriate questions on this test. Let me begin my analysis, where it will become very clear why I have this bad opinion about this question.
I will arrange the equations given as follows and I will begin to solve them in that order:
When I began solving this puzzle, I was skeptical. Part of me said that this would be a question that would be solvable with reasonable assumptions and part of me said that this would be a question that needed unreasonable guesses. In hindsight, I cannot believe how right my latter part was and actually this is also an understatement.
I began with trying to guess the first three equations, i.e. those that had numerical values on their right side. Since 23 and 29 are prime numbers, I assumed that symbols next to each other were simply added (as opposed to multiplied), and, in hindsight, I was correct.
I tried to give values to the symbols in the first equation, so that the second equation would hold.
So, I came up with the following for the first equation:
2 + 3 + 3 + 3 + 3 + 4 = 18 or 4 + 3 + 3 + 3 + 3 + 2 = 18
so that the components of the second equation would be like:
3, 2^4=16 or 4^2=16, 2*2=4 or 2^2=4
and the second equation would be 3 + 16 + 4 = 23.
Things were looking up and I had options. Unfortunately, none of my options seemed to satisfy the third equation. I struggled a lot, until I discovered that the only way to satisfy all first three equations was to have GreenCircle = 2, RedSquare = 3, YellowTriangle = 4 and, and here is the kicker, BaseYellowSquare representing the raise to the second power.
So, when the yellow triangle is alone as a symbol, it is equal to 4. When it is under another symbol, it raises the value of the symbol to the power of 2. And when a yellow square is underneath another yellow square, the yellow square which is above represents the number 4, whereas the yellow square which is underneath raises said 4 to the second power.
For the yellow square to have two different interpretations, depending on whether it was alone (or on the very top of a structure) and whether it was used as base with a symbol on top, is absurd. But since it was the only thing that could make the first three equations working, I went along with it, half-heartedly and with doubt, because of Occam’s razor. “Surely there must be a more sensible interpretation”, I was thinking to myself. In hindsight, things were to become way more absurd, to the final point of complete absurdity. You will see why later on.
So, at this point, things look as follows:
First equation: 2 + 3 + 3 + 3 + 3 + 4 = 18.
Second equation: 3 + 2^2 + 4^2 = 23.
Third equation: 3^2 + 4^2 + 2 + 2 = 29.
From this point, the fourth and fifth equations can be studied independently. From the fourth equation we can get the value of BlueRhombus and from the fifth equation we can understand what the BaseGreenCircle represents. Because, if we are to learn something from our previous experience, symbols here, when used as a base, represent operations rather than numbers.
Let us continue by studying the fourth equation. Let x denote the value of BlueRhombus. We have:
x^2 + (x+3)^2 + 3 = (x+4)^2 + 3^2 + 2 =>
=> x^2 + x^2 + 2*3*x + 3^2 + 3 = x^2 + 2*4*x + 4^2 + 9 + 2 =>
=> x^2 + 6*x + 12 = 8*x + 27 =>
=> x^2 – 2*x – 15 = 0 =>
=> x = -3 or x = 5, and I keep the positive value x = 5.
Thus BlueRhombus = 5. Nothing absurd here. We are at a very sensible point.
Let us continue by studying the fifth equation. The right side is equal to:
3^2 + 4 + 4 + 2 = 19. Now let us study the left side. There is a green circle with 3 red squares on top and a yellow triangle with another yellow triangle on top and on top of the second yellow triangle there is a green circle.
Obviously, the yellow triangle structure is (2^2)^2 = 4^2 = 16. This is because the triangle that is on top along with the circle that is on top of it amount to 2^2 = 4. So the bottom triangle contains the value 4, thus it amounts to 4^2 = 16.
So, if the triangle composite is 16, this leaves the green square structure to be 3. But the 3 red squares that are on top are equal to 3 + 3 + 3 = 9. So, having a green circle as base corresponds to calculating the square root. The square root of 9 is 3 and the green circle with the 3 red squares is equal to 3, because the 3 red squares amount to 9 and 3 is the square root of nine.
Thus, BaseGreenCircle corresponds to the square root calculation.
So, the absurdity continues. When a green circle is used alone (or on the very top of a structure) it is a 2 but it is used as a base it is the square root calculation.
What I found most absurd in the above was that the square root calculation had entered the picture. I was unsure and uneasy with my interpretation, since I believed that the test was about judging intelligence and not mathematical knowledge. And I consider the square root to be a more advanced concept that multiplication or raising to a power. But all evidence pointed to the fact that indeed the BaseGreenSquare was the operation of the square root calculation.
So, I accepted all the above with some uncertainty and skepticism. Little did I know that this was just the tip of the iceberg.
So, we are the following point:
It seems that we are close to the end, and indeed we are, but we are in for a big surprise. Before we proceed, let us recapitulate:
BaseYellowTriangle produces the power of 2 of whatever is on top of it.
BaseGreenSquare produces the square root of whatever is on top of it.
I could not find anything more simple to interpret the first five equations, so I was content that I was near the end. I just had to interpret the sixth and seventh equation and then I would have to calculate the eight expression. To interpret the sixth and seventh equation meant to interpret what the BaseBlueRhombus meant, i.e. what operation the blue rhombus performed when it was used as a base for something.
This is because both the sixth and the seventh equation feature the blue rhombus as a base in their left side.
Let us study the sixth equation. The second part is 5. The first part is the blue rhombus as the base and a blue rhombus on top. So we need to find what operation, let us call this operation f, the blue rhombus denotes. From the sixth equation we have that this operation transforms a 5 to a 5.
Let us now study the seventh equation. The right part is equal to:
4^2 + SQRT(4) + 3 = 16 +2 + 3 = 21.
The left part is a blue rhombus with two yellow triangles on top. Tow yellow triangles are equal to 4 + 4 =8. So, from the seventh operation we have that a blue rhombus transforms an 8 to 21.
Let us recapitulate. We are trying to find the mathematical operation f that BaseBlueRhomus corresponds to. And we have that: f(5) = 5 and f(8) = 21. Once we know what f is, it will be straightforward to calculate the eighth and final expression. This is because the left side of the eighth equation is f(f5+2)) = f(f(7)).
This is easy to see. The left side of the eighth equation is a blue rhombus with another blue rhombus on top and the second blue rhombus has a blue rhombus and a green circle on top. The blue rhombus and the green circle on top are equal to 5 + 2 = 7. So the blue rhombus that is on top is equal to f(7) and the whole structure is equal to f(f(7)).
So, we are the following point:
So, what is f? What is this mathematical operation that sends a 5 to a 5 and an 8 to a 21? From the moment I encountered this puzzle, I thought of the Fibonacci numbers. (Here indeed, 5 and 21 belong to the Fibonacci number sequence.) But the moment I thought of the Fibonacci numbers, I dismissed them. Here in Greece, this is an advanced concept that students do not learn as part of their mathematical education. Here in Greece, I guess that concepts such as the Golden Ratio, Fibonacci numbers and the squaring of the circle are considered esoteric, advanced, unscientific and they are not taught as part of the curriculum. So, my instinct was to avoid Fibonacci numbers, because they are way too advanced for a puzzle that measures intelligence by balancing equations.
Let me explain this a little bit further, so there are no misunderstandings. I am the only one that thinks pi, the Fibonacci number sequence, numbers in general are mystical, esoteric entities. Scientists scoff at notions like mine. Although I treat the mathematical concepts as divine, all others have completely demystified them. I agree with their interpretations, although I think there is something more hidden underneath all our understanding. Anyway, although I find concepts such as those to be esoteric, I vouch for them. This is in contrast with main science which finds these concepts to either be non-esoteric. Because the Fibonacci sequence is a concept that some people like me believe to be esoteric, the Greek curriculum avoids it. Or this is what I think that the reason is.
So, I ignored the Fibonacci number sequence and all other esoteric mathematical entities and I tried to find a mathematical operation that turned a 5 into a 5 and an 8 into a 21.
I thought about decimals, fractions, factorials, but I sort of focused with modulo arithmetic and mostly division. What if this strange elusive operation f was modulo division 5? After all, 21 = 2*8+5, thus 21 mod 2 = 5. I tried desperately to make a 5 go to a 5 and an 8 go to 21 with all sorts of modulo operations, but I couldn’t find any that would persuade me that this is what the authors had in mind.
Occam’s razor. Again. Always.
So, I am back to the Fibonacci sequence. What was Sherlock Holmes saying? “Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.” Then, that is that. I made sure to eliminate all other mathematical operations. But I was still uneasy. The Fibonacci sequence is not an operation. It is a number sequence.
Let me give you the final part of the solution and here you will understand why I am still uneasy.
So, the Fibonacci sequence is produced by starting with 0 and 1 and from then on each number in the sequence is the sum of the previous numbers in the sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, …
If we assign an index of 0 to the first element, then for the function f that supplies a number for the sequence, we have:
f(0)=0, f(1)=1, f(2)=1, f(3)=2, f(4)=3, f(5)=5, f(6)=8, f(7)=13, f(8)=21, f(9)=34, f(10)=55, f(11)=89, f(12)=144, f(13)=233, f(14)=377, f(15)=610, f(16)=987, f(17)=1597, f(18)=2584, f(19)=4181, f(20)=6765, …
So, from the above, we have that:
f(5)=5, thus the sixth equation holds
f(8)=21, thus the seventh equation holds.
So let us compute the eight expression:
Thus, the answer is 233.
Now, what makes me uneasy is that the Fibonacci sequence is not an operation. OK, I admit that if we want to stress things, an operation can be seen as a function and here we have a function f that returns a number from the Fibonacci sequence.
But this is a puzzle concerning the balancing of symbol equations. Only a crazy person would assign the symbol of a blue rhombus to be the function of the Fibonacci sequence and the symbol that is on top of it to be the index of the Fibonacci sequence. You see, I have seen many puzzles concerning the balancing of symbol equations and to me this is crazy thinking.
This leads me to another point I would like to make. I believe that this test is highly inappropriate, because it wants you to be knowledgeable and it wants you to guess. The latter is way worse than the first. The mentality of this test is: “Guess what I am thinking and we will see if you are lucky enough”. And also “I know of this mathematical theory ad discovery and paper that if you too have happened to come across, then you will be lucky enough to be able to answer this question”.
The whole morality of this text is off. Also, I found things there that made me question my sanity.
So, here what I will assume. I assume that this test was put forth to see who objects. I imagine the authors being somewhere shaking their heads in contempt. “No one opposed it!” they will cry in dismay. “No one had the decency to object! Where has this world gone to! Only one person had the integrity to question us. And this person is Dimitrios Kalemis.”
Of course, I do not really believe the above, that this test was put forth to fool us and test us, but it would be the only sensible explanation for me.
I believe that the authors wanted to find highly intelligent individuals to help each other towards a good purpose. I do not know. But this is what I believe. The problem is that the authors claimed to have created a test that measures high intelligence, whereas I disagree.
But the main issue is the following and it reminds me of the P vs NP problem:
Suppose I pose a question that you understand. And suppose that you devote time and energy to try to answer this question. And suppose that you fail to answer this question. And suppose that the answer to this question is an answer than when I give it to you, you will understand it. Then, I say that it is my moral obligation to give you the answer.
The authors of the test do not provide the answers to this test.
If the answers to this test can be understood, then my claim is that the authors have the moral obligation to provide them.
Since the answer of a question is not available, and I happen to know it, I have the moral obligation to provide it.
I am scientist and scientists publish the results of their research. And I just published here my research and these are my results. And this is how the question is answered and this is what it means for the validity of the test.