## My attempt at Question 6 from the Haselbauer-Dickheiser Test

The Haselbauer-Dickheiser Test can be found at http://matrix67.com/iqtest/.

In this blog post, I will study Question 6 from this test.

The question is about physicists’ names as strings. Particularly, each alphabet letter is assigned a number and we have to find which number each letter has assigned to it, given the sums of strings, where each string is the surname of a famous physicist.

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 6.

I have not solved this question, but I promise you, dear reader, that if you bare with me, I will make this blog post worthwhile for you to read and I will provide interesting information about this question.

This is a question that is really hideous and I am against it. It is a very inappropriate question. The question makes you think that you need  to solve a system of linear equations and this is true to some extent. But if you look carefully, FEYNMAN has the letter Y in his name, and the letter Y does not appear anywhere else in the names given. What is worse, the same happens for the letters J, Q, X. Thus, four letters (J,Q,X,Y) do not appear in the 25 names for which we are given the sum of the letters that comprise them.

So, the question is about finding the value of each letter, but the value cannot be solely derived from the sums. We also have to guess. And this is what makes this question inappropriate.

First of all, we have 25 equations with 22 unknowns. The 25 equations are the following:

r+o+2*e+2*n+t+g=104
l+o+r+e+n+t+z=102
c+u+r+i+e=69
m+i+c+h+e+l+s+o+n=109
l+i+2*p+m+a+n=88
m+a+r+c+o+n+i=90
k+a+m+e+r+l+i+n+g+h=120
p+l+a+n+c+k=96
s+t+a+r+k=60
2*e+2*i+2*n+s+t=94
b+o+h+r=47
m+2*i+2*l+k+a+n=103
s+i+e+g+b+a+h+n=87
p+e+2*r+i+n=98
2*r+i+c+h+a+d+s+o+n=155
h+3*e+i+s+n+b+r+g=118
s+c+h+2*r+o+d+i+n+g+e=168
2*c+h+a+d+w+i+k=114
a+n+d+e+r+s+o+n=127
d+a+v+i+2*s+o+n=103
f+e+r+m+i=57
s+t+e+r+n=70
b+l+o+c+k=73
z+2*e+r+n+i+k=99
c+h+2*e+r+n+k+o+v=109

And the 22 unknowns are the following:

a
b
c
d
e
f
g
h
i
k
l
m
n
o
p
r
s
t
u
v
w
z

Please note that the letters j, q, x, y are missing.

I pasted the 25 equations and the 22 unknowns in a form on the site https://quickmath.com. Specifically, I used the Equations -> Solve -> Advanced page https://quickmath.com/webMathematica3/quickmath/equations/solve/advanced.jsp as shown below: Immediately the site produced the following three images. The first image contains the 25 equations: The second image contains the 22 unknowns: The third image contains the solution to this system of linear equations: Below I present the above image in text form, in case you need to copy it.

a = 6
b = 11
c = 20
d = 24
e = 7
f = 16
g = 12
h = 5
i = 9
k = 13
l = 21
m = 2
n = 22
o = 8
p = 14
r = 23
s = 15
t = 3
u = 10
v = 4
w = 17
z = 18

Below you can see how I used Excel to check the validity of the results. The results are correct. The value for FEYNMAN is calculated by setting Y=0, since we do not obtain its value from solving the system of linear equations. Below you can see the results listed in Excel, first sorted by letter and then sorted by number. Below are the same results, in text form, should you need to copy them.

? M T V H A E O I U B G K P S F W Z ? C L N R D ? ?

06 11 20 24 07 16 12 05 09 ? 13 21 02 22 08 14 ? 23 15 03 10 04 17 ? ? 18

We can see that solving the system of linear equations gives us the values of 22 letters of the alphabet, and we miss the values for the letters j, q, x, y. If we assume that each letter is assigned a different value from 1 to 26, then the missing numbers are 1, 19, 25, 26. The previous assumption, that each latter is assigned a different value from 1 to 26, should be valid.

But what are the values of the missing letters j,q,x,y? Which is 1, which is 19, which is 25 and which is 26?

I do not know. We have to guess. We can either consider the pattern of the letters (sorted by their corresponding number) ? M T V H A E O I U B G K P S F W Z ? C L N R D ? ?, the pattern of the numbers (sorted by their corresponding letter) 6 11 20 24 07 16 12 05 09 ? 13 21 02 22 08 14 ? 23 15 03 10 04 17 ? ? 18, or the relationship between each letter and its corrsesponding number. Any pattern that might be found, might be the pattern that was used for the assignments. Another psossibility might be that a clever hash function was used, a function that provides a rearrangement of the numbers from 1 to 26.

Anything goes. But to assume is not a mature thing to ask from someone. Actually, I believe that most people who solved this question gamed the system. They did not really find the pattern of number assignment and instead tried all four possibilities, one by one, to see which one the system accepted as correct. And if they were ever asked how come they proposed their solution, they would just say the truth, or they would just provide a tongue-in-cheek answer that they would know that it would be deemed incorrect.

So the four possibilities are:
Y=1 => FEYNMAN = 75 + 1 = 76.
Y=19 => FEYNMAN = 75 + 19 = 94.
Y=25 => FEYNMAN = 75 + 25 = 100.
Y=26 => FEYNMAN = 75 + 26 = 101.

I believe that the whole point of this question is for us to find the pattern that was used for the assigment of numbers to the alphabet letters. This is why I skimmed throught the solution of the system of linear equations. Indeed one may spend time and reduce the system of the 25 equations to smaller systems and find clever ways to solve it. Or one may solve it manually. Or one may create a program that solves it. But this is not what the question is about. The question is about guessing the value of the letter Y after we have found the values of the 22 letters. And, to me, this is an inappropriate thing to ask.

Update, April 2, 2021: A neat way to assign the remaining letters to numbers might be what I will now describe.

I assume that we have an one-to-one correspondence between the numbers from 01 to 26 and the letters of the alphabet.

We can assume the default order for either the numbers (01, 02, …, 26) or the letters (A, B, …, Z).

Wikipedia has a nice article on permutations, their notation, and the cycles of a permutation. Here is the link:

https://en.wikipedia.org/wiki/Permutation

Taking the above under consideration, I will now present an analysis of both the permutation of the numbers and the permuation of the letters that occurs from the system of equations that we solved. For both the numbers and the letters, I will present a two-line notation for the corresponding permutation, following with the depiction of the cycles that occur.

For the permutation of the numbers, we have the following analysis:

``````Two-line notation:

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
06 11 20 24 07 16 12 05 09  ? 13 21 02 22 08 14  ? 23 15 03 10 04 17  ?  ? 18

Missing numbers: 01 19 25 26

Cycles of the permutation, from shorter to longer:

09 -> 09

03 -> 20 -> 03

02 -> 11 -> 13 -> 02

26 -> 18 -> 23 -> 17 -> ?

01 -> 06 -> 16 -> 14 -> 22 -> 04 -> 24 -> ?

19 -> 15 -> 08 -> 05 -> 07 -> 12 -> 21 -> 10 -> ?

All numbers appear in the cycles, including the numbers 01 19 26.
The only number missing is the number 25.
``````

For the permutation of the letters, we have the following analysis:

``````Two-line notation:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
? M T V H A E O I U B G K P S F W Z ? C L N R D ? ?

Missing letters: J Q X Y

Cycles of the permutation, from shorter to longer:

I -> I

C -> T -> C

B -> M -> K -> B

Q -> W -> R -> Z -> ?

X -> D -> V -> N -> P -> F -> A -> ?

J -> U -> L -> G -> E -> H -> O -> S -> ?

All letters appear in the cycles, including the letters J Q X.
The only letter missing is the letter Y.``````

From the get-go, we knew that the numbers 01 19 25 26 and the letters J Q X Y are missing.

But any one of the four numbers can correspond to any one of the four letters. The analysis of the permutation does not guarantee anything more. We can create any one-to-one correspondence between these 4 numbers and 4 letters and the full 26-element permutation analysis will be updated accordingly.

What I mean is that for any one-to-one correspondence between 01 19 25 26 and the letters J Q X Y, we will end up with a two-line notation and list of cycles. Now, one correspondence might, for example, change the number of cycles, so that the three incomplete cycles might end up as only one very long cycle.

So, the actual 26-element correspondence determines the cycles, and not the other way around. But I had the idea that there is one one-to-one correspondence between 01 19 25 26 and the letters J Q X Y that ends up in a neat analysis. This is of, course, quite subjective, to say the least. Thus, I do not consider this a robust argument. Again, it is the permutation that should dectate the analysis and the number and compostion of the cycles, and not the other way around. But, judging from the way the cycles are composed thus far, it only takes a small step to arrive at the following final configuations.

For the permutation of the numbers:

``````01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
06 11 20 24 07 16 12 05 09 19 13 21 02 22 08 14 25 23 15 03 10 04 17 01 26 18

Missing numbers: 01 19 25 26

09 -> 09

03 -> 20 -> 03

02 -> 11 -> 13 -> 02

26 -> 18 -> 23 -> 17 -> 25 -> 26

01 -> 06 -> 16 -> 14 -> 22 -> 04 -> 24 -> 01

19 -> 15 -> 08 -> 05 -> 07 -> 12 -> 21 -> 10 -> 19``````

For the permutation of the letters:

``````A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
X M T V H A E O I U B G K P S F W Z J C L N R D Q Y

Missing letters: J Q X Y

I -> I

C -> T -> C

B -> M -> K -> B

Q -> W -> R -> Z -> Y -> Q

X -> D -> V -> N -> P -> F -> A -> X

J -> U -> L -> G -> E -> H -> O -> S -> J``````

As we can see from the above, the number 25 goes into the 4th cycle and we just end each of the last three cycles without connecting them with each other. Thus, the proposed correspondence between the numbers 01 19 25 26 and the letters J Q X Y is:

``````J -> 19
Q -> 25
X -> 01
Y -> 26``````

Again, we need not have these cycles. We can create any other one-to-one correspondence between the numbers 01 19 25 26 and the letters J Q X Y, and then study the resulting analysis for the full 26-element permutation. It may not be as neat as the one presented above, but, then again, this is not a definitive argument. 