The Haselbauer-Dickheiser Test can be found at http://matrix67.com/iqtest/.
In this blog post, I will study Question 22 from this test.
The question gives a 10 X 10 square grid. Each square has a color and a single digit number. Also, a number appears after each row and each column, except after the third row, where there is a question mark. The question makes obvious that we have to find the number that goes in the place of the question mark.
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 22.
When I first encountered the question, I saw the numbers outside the grid and they looked like sums to me. They looked like a the result of a calculation that involved the corresponding row or column. In hindsight, yes, I was correct. But the question does not state this. You have to guess. Guessing is not a good thing. Guessing is a highly inappropriate thing. I am talking in the context of an IQ test. It is so easy for me to create an IQ test like that. I will name it “Guess What I Am Thinking!”. And you will have to guess whatever rule I coined. That’s not right.
So, as it turns out, these numbers are sums. But if they are sums, is the question ill-conceived? And does the question allow you to cheat?
What I mean is that if these numbers are sums and each square is summed depending on its color and its number, then the sums of the column sums will be the same as the sum of the row sums.
In other words, we should have:
95 + 93 + 62 + 77 + 106 + 100 + 102 + 100 + 78+ 97 =
= 80 + 92 + x + 89 + 98 + 91 + 78 + 99 + 83 + 88
And from the above equation, we have x = 112.
Indeed, in hindsight, this is the correct answer. In this blog post I will explain why I am certain that this is the correct answer.
But I do not think that the examiner wanted the solution to be found this way. Again I am guessing. There is and will be a lot of guessing involved in this test. And this is one of the things that is bad about it.
So the correct answer to life, the universe and everything, or just the missing number in this question is 112. But if the test was administered in an environment where you could type your answer and receive the result of whether your answer was true or false, you could easily game the system and try 112 or other different numbers and then, when you would have been informed that your answer was correct, you could use this information to trace things backwards.
So, one of the problems pertains to the way the test is administered. Answering 112 does not amount to much. You have to state why you gave the answer that you gave. But I am afraid that the test is not administered in this way, but it is administered in a way that allows you to cheat and game the system.
OK. Let us suppose that we are not certain that the numbers outside the grid are sums and we are not certain that the sum of the numbers in the bottom is equal to the sum of the numbers on the right. How will we proceed to answer this question?
Here is what we have to do. We will assume that each number outside the grid is the sum of the numbers in the corresponding row or column. But each number will have to be differentiated according to the color of the square that it resides on.
Any normal person would assume that each color corresponds to a number, and in hindsight, this is correct. Any normal person would also assume that the number of the color multiplies the number of the digit. But in hindsight, this is not correct.
Anyway, I did not know this and I assumed that the number corresponding to the color in each square multiplied the digit in the square. So I had to find the number that corresponded to each color.
Here is what I did. I observed and observed and observed the grid. And I noticed that the sixth column from the left contains only four colors (pink, green, blue, red) of the five colors in the grid (pink, green, blue, red, yellow). Specifically, in the sixth column we have the following:
The sum of the red squares is 9 + 2 + 6 + 4 = 21.
The sum of the pink squares is 5 + 3 = 8.
The sum of the green squares is 4.
The sum of the blue squares is 4 + 8 + 4 = 16.
And the sum (I loosely refer to it as sum, it is really the result of an unknown calculation) of the whole column is 100.
Since I was assuming that each color multiplies the digit by a constant number, I immediately knew that the red color corresponded to an even number. Why? Because the sum was an even number (100) and the sums from the other colors each was an even number. So, even if their color corresponded to an odd number, the multiplication would end up in an even number. Since the whole sum had to be even, it was without doubt that the red color was an even number.
So, red had to be 2 or 4, because had it been 6 or higher, 21 * red would be more than 100. Actually, red could only be 2, because if red was 4, then even if the other colors were each 1, the sum would end up more than 100. Also, each color had to be a different number, because it would be illogical not to.
So, it should be that red =2 and pink, green and blue had to have different values.
The problem was that the equation
21 * red + 8 * pink + 4 * green + 16 * blue = 100 =>
=> 21 * 2 + 8 * pink + 4 * green + 16 * blue = 100 =>
=> 42 + 8 * pink + 4 * green + 16 * blue = 100 =>
=> 8 * pink + 4 * green + 16 * blue = 58
does not hold for any integer values we may try for pink, green and blue, given that we have to choose from the integers 1,3,4,… and all three should be different. (I remind you that 2 is missing, since red =2).
So, I struggled with this for a while, until I gave up. So, it seemed that the multiplication concept was not what was going on in the calculation. So, how where the “sums” produced? I thought that if there was not a multiplication of the color number, it would be an addition of the color number.
So, I assumed that each color corresponded to a different integer and that integer was added to the digit that was in the square. And, in hindsight, that was the correct assumption.
So, I had to test my assumption. I had to assign a different integer number to each one of the five colors and calculate the sum of each column and row, where for each square I would add the color number to the digit.
But I had to do a what-if analysis. I wanted to try different values for each color to get the sums to match those given. So, here is what I did: I used Excel.
I created five names, one for each color, and I would set integers as their values, to see if the sums would match those that the question gave.
Each cell was the sum off the digit and the color. The digit was a number and the color was an Excel name. So, the cell A1 would have the formula =3+red. The cell B1 would have the formula =4+green. And so on.
In hindsight, it would have been easier for me to do all the checks I made, if I did not use names but instead I would use cells. This is because I had to open name manager each time I wanted to make a change, whereas a cell would always remain open in front of me, thus making any changes and what-if analysis easier.
Anyway, I struggled a little bit, trying different values, always being careful to use different values for different colors and doing the analysis primarily on columns which contained fewer than the five colors. Whenever I got a sum that was correct in one row or column, another sum would be off, so I struggled a little bit, I tried different permutations of integers and I finally got one that produced all sums correctly. The is depicted in the following screenshot.
I am certain that this is the correct answer, since it would be highly improbable for the examiner to have something else in mind. Each row and column sum is validated and checked to be in agreement with what the question provides.
So now we know that the numbers outside the grid are the sums of each row and column, where by sum we mean the calculation that takes each square and adds the color to the digit and then adds all these sums. Blue = 7, Green = 2, Pink = 6, Red = 4, Yellow = 3. The missing number is 112.