My attempt at Question 4 from the Haselbauer-Dickheiser Test

The Haselbauer-Dickheiser Test can be found at

In this blog post, I will study Question 4 from this test.

The question is about spheres stacked up to create tetrahedral pyramids.

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 4.

This question is great, if you plan to submit it to an elementary school student. But for a test for high intelligence, it is ridiculous. Not only that, but this question (question 4) and another question from the test concerning spheres (question 15) are asked and answered in Martin Gardner’s book titled “New Mathematical Diversions”.

These are not the only questions that are blatantly copied word for word from Martin Gardner’s books, but here the situation becomes extreme: Two questions, from the same book and also …  wait for it… from the same chapter: chapter 7, titled “Packing Spheres”. What more can I say? All I can ask is: Are the people who created this test completely deranged?

I do not know where to begin. First of all, the answer is really easy to find. I would pose this question to an elementary school student. (By the way, speaking about “school”, who came up with this invention? Some invention they made. Give me the person who invented school and I will sent her to a group of students to lynch her.)

Before I present the answer I came up for this question, I would like to present the pages from Martin Gardner’s book where this question and the corresponding answer are mentioned. I promise that my next blog post will be about the other question from the test that is asked and answered in the same chapter.

So, question 4 from the test can be found in page 84:

and the corresponding answer can be found in page 90:

And yes, underneath the answer for this question, the answer for the other question is also found. One page, page 90, contains the answers to two questions from the test. And, yes, the creators of the test want to preserve the integrity of the test. And yes, all the above make no sense at all.

I will now present an answer to this question, by thinking as an elementary school student would. With simple thinking and the use of Excel, I found the answer in a few minutes.

I opened Excel. I created three columns with numbers. The first column had the natural numbers 1, 2, 3, … that denote the layer of a tetrahedral pyramid starting from the top. The second column had the number of spheres that comprise the layer. This number is equal to n(n+1)/2, where n is the number of the layer. The third column is the running total of the second column, thus it is the number of spheres of the layer and all the layers above it. Thus , the third column depicts the total number of spheres that a tetrahedral pyramid contains.

Now, the problem we want to solve is to take the numbers from the third column and, from these numbers, find the minimum number that is the sum of two others. In order to do that, I put the numbers in a horizontal line and in a vertical line, thus forming a square. For each pair of numbers, I had Excel calculate their sum. Then, all that was left to do, was to search in these sums to find a number from the third column.

I needed to only search either above or below the diagonal, since the sums below the diagonal are the same as those above the diagonal. Searching is easy because the value of the numbers is increasing from top to bottom and from left to right.

So, I found that the answer is 680.

A tetrahedral pyramid with 8 layers contains 120 spheres. A tetrahedral pyramid with 14 layers contains 560 spheres. A tetrahedral pyramid with 15 layers contains 680 spheres. 120+560=680 and 680 is the smallest number for which this occurs. That is, for which the sum of the spheres of two different tetrahedral pyramids is equal to the number of spheres of another tetrahedral pyramid.



About Dimitrios Kalemis

I am a systems engineer specializing in Microsoft products and technologies. I am also an author. Please visit my blog to see the blog posts I have written, the books I have written and the applications I have created. I definitely recommend my blog posts under the category "Management", all my books and all my applications. I believe that you will find them interesting and useful. I am in the process of writing more blog posts and books, so please visit my blog from time to time to see what I come up with next. I am also active on other sites; links to those you can find in the "About me" page of my blog.
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