The Haselbauer-Dickheiser Test can be found at http://matrix67.com/iqtest/.
In this blog post, I will study Question 3 from this test.
The question is about the maximum number of regions that a circle can be divided if we draw chords connecting points on its circumference.
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 3.
Well, any question from this test leaves me speechless, but this is not a compliment to the test. I find that the test is pointless, tedious, unintelligent and lacks explanation. I cannot get it. I cannot understand it. I do not know what to say. I am speechless.
Here we have this question. Question number 3. What does it want from us? Well, it is very clear what it wants. I wants us to find the maximum number of regions that are created when there are seven points on the circumference of a circle and we connect them with straight lines.
It is a nice problem and a known one, with a known solution. Since the authors of the test want it to maintain its integrity, why pose a well known problem? Points for originality: Nada! Since the authors of the test want to test intelligence, why ask a question that in this blog post I will prove that a toddler can answer? Points for intelligence: Nada! What is the purpose of such a question in a test for high intelligence? Points for sense and sensibility: Nada!
When I try to solve a question from the test, I feel to encounter two S’s. Solitude and Stupidity. Am I the Sole person who thinks that this test is Stupid?
Before we go any further, I would like to repeat that is a well known problem with a well known solution.
I do not where to begin, because this problem can be solved by a toddler, but more on this later.
First of all, this problem is discussed in the Martin Gardner’s book “Mathematical Circus”. Below I present the pages of this book that pertain to this problem. Also, at the end of the book there are references pertaining to this problem.
There are quite a few questions from this exam that are depicted almost word-for-word in Martin Gardner’s books. As I said: Points for originality: Nada! I will let the reader decide who is the unoriginal: Martin Gardner or the authors of the test. At least, Martin Gardner names his sources!
So, here is the material from Martin Gardner’s book.
Below, I present two links that provide the solution (and the rationale behind it) to this well-known and well-studied problem.
But this problem can be solved by a toddler as well. The toddler may not be able to provide a general formula, but she will be able to count the number of regions. To prove to myself that a toddler could do it, I put myself in a toddler’s position. Although I did not crawl on all fours, (do not be surprised, that would be ridiculous for anyone else, but not for me!), I drew the relevant shape and counted the numbers of regions. Below, I present all cases from 2 points on the circumference up to 7 points on the circumference. Obviously, since we want the maximum number of regions, care has to be taken that each crossing point is made by only 2 lines. If 3 or more lines pass through a crossing point, this means that 1 or more regions are collapsed to this point, so we do not have the maximum number of regions.
I drew each shape in PowerPoint. As I drew each line (chord), I would count it, so that I knew how many lines there were, even if the question did not ask for it. Then I opened each image in Paint. I took the area painter (the bucket) and started clicking inside each region, thus counting them. Each region I would click and count was colored by my click, so I knew not to double count it.
And I did not stop there. When I had counted the regions by coloring them with the same color, in the colored shape the lines and the intersection points were still shown. So I took the eraser and I put a little erasing smudge on each intersecting point, thus counting the intersecting points as well, even though the question did not ask for it.
I created the following table which depicts my results. “Points” are the number of points on the circumference. “Chords” are the line segments, where each line segment is drawn between two of the points in the circumference. “Crossings” are the inner crossings, the crossings where two line segments cross inside the circle. “Regions” are the empty areas that are formed inside the circle, “empty” meaning they contain no lines inside them.
The above table shows that the answer to the puzzle’s question is 57.
Let us see how we can generalize our findings. The following analysis is widely known to all mathematicians and also physicists like me know this theory very well. Dare I say, all students know this theory very well? But this measures knowledge or intelligence? I say that it only measures knowledge. Anyway, the test question only asks for the number of regions, so the answer is 57 and that is all. By merely counting the regions we arrived at the answer.
First of all, let us denote the number of points on the circumference with the letter n. This is the column named “Points”.
In order to have the maximum number of regions, each point is connected to each other point with a chord. To find the total number of chords, we can think in one of three ways.
First way: We have n points. Each point is connected with each one of the other n-1 points. So we have n*(n-1) lines (line segments). But in this number, we have accounted for each line twice, from the perspective of both points in each line. So we have n*(n-1)/2 lines.
Second way: From the theory of combinations. The number of lines is
(n choose 2) = n!/[2!*(n-2)!] = n*(n-1)/2.
Third way: We can say that the first point made n-1 connections with the other points and left the building. Then the second point made n-2 connections with the remaining points and left the building. And so on, until the second from last point made 1 connection with the last point at left the building. Thus, the last point is now alone and can make no connection. Thus, the number of connections (lines) made is
(n – 1) + (n – 2) + … + 2 + 1 = n*(n-1)/2.
So, the formula for the column named “Chords” is (n choose 2) = n*(n-1)/2.
Now let us find the formula for the column named “Crossings”.
For 2 lines to create a crossing, the 2 points of one line have to be different than the 2 points of the other line. In other words, to have a crossing, 4 different points on the circumference have to be involved, two of them defining one of the lines and the other two defining the other line.
This concept is easier to grasp if you look at the shape I provide for the simplest case. This is the shape with 4 points on the circumference which corresponds to only 1 crossing. This sole crossing exists because the 4 points on the circumference are 4 different points. Any 4 different points define two lines and 1 crossing.
Thus, the formula for the column named “Crossings” is (n choose 4) = n!/[4!*(n-4)!].
Now let us find the formula for the column named “Regions”, which is what the puzzle describes as regions.
The Wikipedia article I mentioned above rigorously describes the different approaches that explain and provide the corresponding formula. Because someone, like me, may lose themselves in mathematical rigor, before reading the full rigorous explanations and proofs, it is best to have a high level overview of them.
So, as an introduction to these rigorous proofs and explanations, I would like to give a simplistic overview.
Imagine (you don’t have to imagine, you can actually draw what I describe) that we have the circle and nothing else. So, we only have 1 region, which comprises the full circle. Then imagine that we draw parallel lines. As each new line is drawn, another region is created. Now imagine that the lines we already drew start intersecting. As each intersection (crossing) occurs, another region is created.
So, the formula for the column named “Regions” is
1 + Chords + Crossings = 1 + (n choose 2) + (n choose 4).
So, I can present you with the following theoretical table which coincides exactly with the empirical table I gave you above.