The Haselbauer-Dickheiser Test can be found at http://matrix67.com/iqtest/.
In this blog post, I will study Question 10 from this test.
The question is about an analogy.
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!
This is one of the most idiotic questions ever posed. But is it the most idiotic question ever posed? Or is there a question that tops it in the idiocity department? The fact that I am contemplating this, does not really vouch for this question. As I said, this is my opinion.
So, a bunch of red shapes corresponds to 58. And, in a corresponding way, a bunch of yellow shapes corresponds to something. What is this something? Probably another number, and probably a two digit one, like 58. Is it 58 or “5” and “8” stringed together? Whatever. We are dealing with idiots, remember?
You can try and come up with ways to relate the number 58 (or the numbers 5 and 8) to the red shapes. After that, you can use the same thinking to relate another number to the yellow shapes. This is not the correct way to approach this problem, but before I talk about the correct way, let me provide you with one (out of infinite many ways) to relate the red shapes to number 58.
If you count the number line segments used to draw the red shapes, you can find a sum of 58 line segments, if you try to count in specific ways. One of these ways is the following:
or the following:
The difference is in the way the third shape’s line segments are counted. But in both cases, they amount to 13 line segments.
These attempts seperate the line segments in a way as to add them up to 58. This line segment separation and count leads to the corresponding way for the yellow shapes, perhaps as follows:
You may add the numbers and get a sum. And say that this is the answer. But it may not be, really. And we can argue which is the most consistent way of counting the yellow shapes’ line segments, so that they relate exactly to the red shape’s line segments counting. But it does not matter, since this may not be the correct way to solve Question 9.
And we can find other ways like that. We must be very careful though. Each way of solving that we suggest, must be unambiguous. It must function in an unambiguous manner. So, let us say that we are counting. The way we are counting must leave no ambiguity. And also, and this is very important, even if a way is specific for the red shapes, it is of no value if it is not specific for the yellow shapes as well. So, if you found a non-ambiguous way to produce 58 from the red shapes, but this way is ambiguous (due to the different case) for the yellow shapes, then your method is not valid.
But these ways, and any others like that, do not seem to be the correct way to proceed. In this blog post, I will explain to you what the correct way to proceed is.
When I was looking at these shapes, I was alert to the fact that we must process all information that these shapes provide. We may not pick and choose. And first of all, we have to understand what these shapes are. Where do they come from? What is their story? What do they depict? Why do the yellow shapes not all have the same length? Why do the yellow shapes have smaller height than the red shapes? What is going on here? Am I on candid camera?
There are 6 red shapes. And there are 6 yellow shapes. Let us focus on the 6 red shapes. Why are there 6 of them? And they all have the same dimensions. Like the 6 sides of a cubic box. But these cannot be the 6 sides of a box. So, what can they be? At some point, I understood exactly that. Do you know what they are? They are projections. They are projections of a cubic solid. A cube has 6 sides, so we have 6 projections: front and back, left and right, top and bottom. 6 if you count correctly. So, this is why we are given 6 red shapes. Each one corresponds to the way the solid appears from this side. Of course, since the solid can be oriented any way in space, naming front as front, is an arbitrary convention. We will just pick a convention, when we get to that point.
The Internet, especially Pinterest, is full of projections like these. Search for orthographic projections, orthometric projections, isometric projections, vistas, or any other relevant term you may find. Among others, I found a very nice link:
https://med.se-todo.com/other/47422/index.html
I took the liberty to copy one of the images from there to here, for your convenience:
So, we consider a 3X3X3 cube, which is built with 1X1X1 cubes as elements. We may remove some of its elements. We want to describe the resulting cubic solid. So, we take three projections: top, front, side. Now, this can be a nice puzzle. Starting from the cubic solid, we want to draw the three projections. Or, starting from the three projections, we want to find the cubic solid. Well, this second case is the one Question 10 is about.
Let me explain how the cubic solid is formed. It is like a Rubik’s cube, if a Rubik’s cube was comprised of 27 cubic elements and it was solid, i.e. no turning sides.
Except in our cubic solid, some of the 27 elements may be missing. So, imagine a 3X3X3 cubic solid, constructed from at most 27 smaller cubes, all identical in dimensions. At most 27. And, as I said, some of the smaller cubes may be missing. So, there is a huge number of possible 3X3X3 cubic solids than can be constructed this way. But, I guess that each one of these cubic solids should be “connected”.
“Connected” is a term that I coined. Or rather, I copied from Graph Theory, a branch of Mathematics that deals with graphs. A graph has nodes and edges (which are lines connecting the nodes). If every node in a graph is connected to at least one other node, then the graph is “connected”. No node left behind. So, in our cubic solid, a smaller cubic element is always connected to at least one other smaller cubic element. Or else, the “shape” will no longer be one shape, but two or more independent ones.
As you see from the example icons I posted above, the projections (or, equivalently, views) are usually 3: front, top, one of the sides. But here, in Question 10, we have 6 projections. So, this got me thinking: We either talking about two cubic solids, or we are talking about one cubic solid, but then, the projections are: front and back, left and right, top and bottom. So I said to myself: I will try to come up with only one cubic solid. If I cannot, I will look for two seperate cubic solids. But I was quick to find out a single cubic solid that corresponded to these 6 projections and I had the problem solved.
Here is how I did it.
I was planning to get physical 1X1X1 cubes to build such 3X3X3 cubic solids, or draw them in some way in my computer in 3D, but I did not need to. After thinking for a little while, I had the problem solved using just a 2D piece of paper.
Here is how I ended up relating the projections to the cubic solid:
I drew the three layers of the cubic solid as follows. Each 1X1X1 element would have a question mark, until I decided whether it existed, in which case I would color it, or it would not exist, in which case I would leave it white by erasing the question mark.
And I imagined that the base layer sits at the bottom, on top of the base layer we place the middle layer, and on top of the middle layer we place the top layer.
Now all, that is left is to decide which of the 27 1X1X1 cubes exist and which do not.
I started with the sides that have the most flatness. Those are the ones form the left, which have the largest uninterrupted areas. Because when lines are involved, this means that we change depth.
So, I took the first projection from the left. I decided to evaluate it as the back side, since I could visualize it without seeing it and since I needed the front unobscured so I could keep working. And I colored the 1x1X1 elements that were needed to have this projection at the back, as follows:
Quite an improvement, don’t you think? Please notice the middle layer. Do you see all the white elements on both sides, having no question marks? This means that they do not exist. This is because the first red projection from the left, which I evaluated in the back, has a thin waste! You can see throught the sides!
Alright, let us proceed. I then examined the second red projection from the left and I evaluated it at the top layer. The previous projection was quite symmetrical to pose any objections! Yes, evaluating it on the top layer, it would obscure things, but I was now feeling very confident. In the top layer, I already had a beginning (three elements drawn red), so I placed it accordingly, as follows:
Please notice that I did not yet place the two-element area. These two red squares are in a different layer (either the base layer or the middle layer).
In order to place these two elments, I considered two things. The first was the bottom side. It had a large 7 element area. The second was the left and right side. They had to be see-through in two middle elements.
And thus, finally, I ended up with the following cubic solid:
This is it. This is the solution. It is a 27 element cubic solid with 15 elements existing and 12 elements non-existing. The middle layer connects the base layer and the top layer with only one element, which is not good for the structural integrity of the whole thing. Not that it matters, just saying.
Ok, let us prove that this is the correct cubic solid. We will examine each of the 6 projections given, to see if we get the correct view and explain the areas of each projection. Here we go:
So, we have the correct cubic solid. All projections (and all individual areas of each projection) are accounted for. I will again present the cubic solid for your convenince:
How is it related to 58? Well, I cannot look inside the sick little minds of those who posed the question. But there are a lot of things that we can consider. I will only provide a brief elementary analysis.
Just by looking at the layers vertically, we see three columns of 5 red (existing) elements each and the largest white (non-existing) number of elements is in the middle layer and their count number is 8.
But let us be a little bit more methodical.
First of all, we have a cubic element with 27 elements, 15 red (exsiting) and 12 white (non-existing). We can study percentages of these, if we wish.
Another analysis that can be done is to list and count the elements, either from base to top, or from left to right, or from back to front. Here is this analysis, which is meant to be read vertically:
These numbers may as well be taken into consideration.
So, the cubic solid or its projections are somehow related to 58. The analysis above may help in finding this relation, which could be virtually anything.
Now that we have finished our analysis about the red projections, let us begin to study the yellow projections.
It is immediatelly obvious that we have to deal with a 3X2X2 rectangular cuboid solid. The cuboid has 3X2X2 = 12 elements, and the middle ones are elongated in length, whereas the outer ones are not. Some of these 12 elements will exist, and some will be missing. And I guess, this yellow cuboid will have to be “connected” somehow, since the red one was too. Below I present what I have in mind, with a view from above looking down.
So, let us find the cuboid in question. In the end, we will have something like the following relation:
We will begin by creating the layers. We can choose to have two 3X2 layers, the base layer and the top layer, and we can imagine the base layer to be at the bottom and the top layer to go at the top. In the beginning, all elements have questionmarks, since we do not yet know if they exist or not.
We will begin by choosing the back side, because it has a large area and this will help. By placing it in the back, we have the following configuration. Please note that we deleted the two questionmarks in the top layer, because these elements are non-existing.
Great! We progressed quite a bit with just one projection. If we take into account all the other porojections, we arrive easily at the following cuboid:
Let us prove that this cuboid is the actual cuboid we are looking for, by deriving the 6 projections from it.
Again I present the 3X2X2 cuboid here, for your convenience.
Let us do a little rudimentary analysis. We have a cuboid and its 6 projections. The cuboid is 3X2X2, so it has 12 elements, of which 6 are existing and 6 are non-existing. I also provide the following analysis, which is meant to be read vertically:
I guess, that is all I have to say. I hope I shone a little light as far as Question 10 is concerned. It is about the 6 projections a cuboid has and that these projections can be used to find it and describe it.
Since my answers are quite lengthy, one might be inclined to assume that the questions of this test may have some merit. This is the point I want to clarify. They do not. The way I see it, I have to make lengthy answers, because the questions are not well posed. In the world I want to live in, these questions would be a book’s length. Then the questions would have merit and the answers would be of reasonable length.
Update, March 15. 2021: I have tried to find how the red cuboid is related to the number 58. I have found some ways, but they do not seem to be robust. They seem to be ambiguous.
But then, after some consideration, I found a way that seems robust and unambiguous enough. This is counting the external tiles of the reb cuboid. In other words, how many squares do the external sides have? If we need to count the external tiles or if we need to cover the cuboid with new tiles, how many tiles would we need? If we were to paint it (with red or any other color), how many tiles would we need to paint?
I counted the external tiles and since the count number is 58, this seems that it is a robust answer.
Let us do this counting for the red cuboid.
We are counting the external tiles.
We begin with the top layer. We have 7 top tiles. We have 6 botton tiles, because the seventh one is hidden between the top and middle layer. We have 16 side tiles.
We continue with the middle layer. We have 4 side tiles. We do not have any top or bottom tiles, since the only top one is hidden between the top and middle layer, and the only bottom one is hidden between the middle and base layer.
We finish with the base layer. We have 6 top tiles, because the seventh one is hidden between the base and middle layer. We have 7 bottom tiles. We have 12 side tiles.
Let us add all these numbers: 7 + 6 + 16 + 4 + 6 + 7 + 12 = 58.
This seems to be the correct relation between the red cuboid and the number 58. So, we are not talking about two seperate numbers, five and eight. We are talking about the number fifty eight, which corresponds to the number of external tiles of the red cuboid.
Since this is the correct relation, we can extrapolate this thinking in order to find the number of external tiles of the yellow cuboid. But here, a problem arises. Not all of the yellow tiles have the same dimensions. The middle ones are elongated.
We will discuss about this, but, first, let us do the counting for the yellow cuboid.
We are counting the external tiles.
We begin with the top layer. We have 2 top tiles. We have 8 side tiles. We have no bottom tiles, since both are hidden between the top layer and the base layer.
We finish with the base layer. We have 2 top tiles, since the other two are hidden between the top layer and the base layer. We have 4 bottom tiles. We have 10 side tiles.
Let us add all these numbers: 2 + 8 + 2 + 4 + 10 = 26.
Now, this is a valid answer. But if we want to account for the 4 elongated tiles around the middle of the base layer, we need to find a way to do it. If we assume that their area is twice the area of a small yellow tile, which is a big “if”, then we need to add 4 more tiles to the count. Thus, we end up with 26 + 4 = 30.
So, I have all but proven that Question 10 is ambiguous, which means that it is open to interpretation. Some of the questions in this test are. For example, another question, Question 8 is also ambiguous, and I have discussed about its ambiguity in my corresponding blog post.
As far as Question 10 is concerned, we can view the yellow cuboid as a 3X2X2 cuboid with the middle elements elongated, in which case the number of external tiles is 26. Or we can view the yellow cuboid as a 4X2X2 cuboid with all elements identical, in which case the number of external tiles is 30. I depict this second case below:
This second case seems to be even more robust, but only if we assume that the middle tiles are twice the length of the other tiles. And, as I said earlier, this is a big “if”.
Let us again count the external tiles for this 4X2X2 cuboid.
We begin with the top layer. We have 2 top tiles. We have 8 side tiles. We have no bottom tiles, since both are hidden between the top layer and the base layer.
We finish with the base layer. We have 3 top tiles, since the other two are hidden between the top layer and the base layer. We have 5 bottom tiles. We have 12 side tiles.
Let us add all these numbers: 2 + 8 + 3 + 5 + 12 = 30.
So, now we have all the data that we need to make an informed evaluation of this question.
Dimitrios Kalemis 