The Haselbauer-Dickheiser Test can be found at http://matrix67.com/iqtest/.
In this blog post, I will study Question 17 from this test.
The question is about cube nets covering an area.
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 17.
This question shows most everything that is wrong and bad about the whole test. I do not know where to begin. Honestly.
Ok, I guess I will have to begin from somewhere. Any start I think of choosing, I can find a worst one. So I guess I will start from the first thing that comes to my mind: discrimination and unfair advantage. This question about cube nets. Hexominoes and cube nets as their subsets are know to elementary school children in the USA. But not in Greece. Here, a child will be lucky if she only encounters and learns about one of the cube nets. But in the USA, children learn about the 11 kinds of cubes nets. They are even given all kinds of hexominoes and asked to discern which ones are cube nets. A child in Greece, nay, any person in Greece has absolutely no notion of what I am writing about. So, how come do I know these things? Well, I am not like other Greeks. Although I was born and raised in Greece, my culture resembles that of an American. Blame it on TV and radio. Blame it on all the scientific books in English I have read throughout my life. Blame it on me having interests way different that other Greeks. Well, blame it on anything you want, my culture has nothing to do with a usual Greek person’s culture. I have listened to the great radio programs from AFRTS one too many times. I have watched American movies and TV series one too many times. I have read American text books one too many times. So, the point I am trying to make is that any normal but also highly intelligent Greek would not understand what this question is about and/or be at a great disadvantage compared to an American.
Another thing that really bothers me is that this question allows you to cheat. The grid has 96 squares. Each cube net has 6 squares. 16 X 6 = 96. So, the least amount of cube nets that would cover the grid with no overlap is 16. So, it is probable that the answer is 16 cube nets. If not, then the test taker can start adding one more to the answer, until she finds the correct answer. In hindsight, and as I will show in the last part of this blog post, the answer is indeed 16, meaning that the grid can be covered with cube nets with no overlap among them.
This really bothers me because one may go ahead and guess that the answer is 16. Is is easy to guess that. Coming up with a configuration that achieves it is a little bit harder. But the test does not demand an actual solution, an actual configuration. It just demands the number. This is unacceptable to me. And, in any event, I cannot consider this question and the corresponding answer as indicators of high intelligence. All they are is a fun way to spent an afternoon. And this is something that unfortunately I cannot say for the rest of the questions of this test, because most of them I consider boring and tedious. In these series of bog posts, I want to prove that this test is detrimental to people and mankind, but I am faced with boredom, since for many of the questions, although I know how to solve them, I find it tedious to get involved with them. I have always been honest with you, dear reader, and I am not willing to change that. So, in all honesty, if I had a girlfriend, I would not have gone to the trouble of solving the questions. I would have devoted my time to said girlfriend. But since I have spare time, I do this, in case you find it helpful. And still, each time I try to tackle a question, I say to myself that this is stupid and pointless and there are way better and more sensible problems I can solve in my spare time.
As for this problem, I previously said that it is a fun way to spent an afternoon. Well, yes but for toddlers. Mental note: Next time toddles are in the house, (toddlers are frequently here, children of friends and neighbors) I would give this puzzle to them to solve it. And some others puzzles from this test. Because, honestly, I am grown person and I should not be involved in this nonsense. Am I certain that some of these questions may pass as legitimate to a child. But if I tell them that this is test that measures high intelligence, I know what they would tell me. I have been around long enough with toddlers to know their come back answer. If I tell them that this is test that measures high intelligence, they will tell me: “I don’t believe you.” And the last thing you want is to lose your credibility with toddlers. They are harsh and unforgiving judges.
I can say a lot more about how bad this question is and how atrocious the whole test is, but let’s cut this short and instead focus on the solution of this particular question. I was talking about hexominoes. They are constructs made from six squares. From all possible hexominoes, 11 are cube nets. You can look up “hexominoes” and you can look up “cube nets”. To help you, I present the following two links:
For your convenience, I present a drawing containing the 11 cube nets.
The question wants us to use these to cover the grid given. We are not allowed to place a cube net that is partly outside the grid, and the word “outside” includes the white squares that are inside the grid. The cube nets must not cover any part of those as well. Each cube net must cover blue squares of the grid and only them. And we want to cover the grid with the least amount of cube nets. If we can achieve no overlap among the placement of the cube nets, then this is the configuration with the least amount of cube nets. And yes, we can achieve such a configuration. And since there are 96 squares in the grid and each cube net has 6 squares, a total if 16 cube nets cover the grid and it is the least number of cube nets that can achieve this.
Let me give you such a configuration and tell you how I came up with it. I prsent my solution below:
No, it is not some satanic drawing of symbols, I just opened the grid in Paint and started drawing cube nets with a yellow color line. I noticed that the grid has a 4-way symmetry. So, if I was able to fill one side with 4 cube nets, then the other 3 sides would be exactly the same. So, I had to find a configuration 0f 4 cube nets which would fit with itself in its adjacent areas, in its borders.
So, what I did is that started placing two cube nets on a side and I would do the exact same arrangement in the other 3 sides. Then I had to be extremely careful so that the third cube net would be such is such a way that it allowed the fourth cube net to cover exactly the remaining space. This was easier said than done, bit I finally came up with the above configuration. I hope you notice the 4-way symmetry. To help you, you can imagine any four cube nets in series. When you repeat this series, each time rotating it 90 degrees, you will fill the entire grid. Each of the four sides of the grid is filled in exactly the same way and with exactly the same cube nets in exactly the same arrangement.
One last thing I would like to point out about unfair advantages is that although I solved this problem with just thinking about it, there are people who are involved with hexomino tiling as scientific research. And computer programs also exist that allow you to specify a grid with forbidden areas and to specify which hexominoes to use then and perform the tilling of the grid for you. Test takers with such knowledge and/or programs are at a huge advantage over other test takers.
Dimitrios Kalemis 