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We continue with our study of triangles in this post. We will consider a few elementary rules related to triangles. I will also give some comments on what we can glean from some of the rules. Here, I wish to apologize to the reader. Typesetting mathematical formulas using HTML is too tedious. Hence, I have…
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In the previous post, Primer to Properties of Triangles, I had solved a geometry problem based on properties of the equilateral and right-angled triangles. I promised that I would use that as a launching pad for a series on Properties of Triangles. Students of mathematics are introduced to triangles and circles quite early in their schooling.…
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After a considerably long break, I resumed with two quite heavy posts on Gödel’s Incompleteness theorems. It’s time to crank down to more reasonable levels of mathematics. I recently came across the problem depicted with the figure below: Point G is where the medians of the triangle ABC intersect. The triangle BDG is equilateral with…
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In the previous post we looked at the system of Gödel numbering that allowed Gödel to map any natural number to a unique mathematical expression. In this post we will look at how Gödel succeeded in using this system of numbering to make mathematical statements refer to themselves. First remember that Gödel used the following…
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I was first introduced to Gödel’s incompleteness theorems when I was doing my master’s degree when one of my apartment mates suggested that I should read Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter. When I read the book, I was immediately hooked. However, to be frank, I did not understand that Hofstadter…
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My annual stint of grading IB papers started this week. So I will tackle only a relatively simple problem today. I will take a break for about three weeks while I am grading papers and resume after that. I recently came across the problem depicted below. We are given that the external quadrilateral is a…
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There are many ill-informed ideas about mathematics. One is that there is a solution to every problem, as suggested by the meme above. This is not true. There are problems that have no solutions. Then there are problems which have solutions, but whose solutions cannot be solved without significant computational power. And there are problems…
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This will be the final post in the series on counting principles. Here, I wish to deal with a strange idea of balancing parentheses. If that seems like a strange term itself, let me explain. An arrangement of n left parentheses ‘(‘ and n right parentheses ‘)’ is said to be balanced if, for every…
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Launching Pad We are in the middle of a series on counting principles. In The Inclusion-Exclusion Principle, we saw a method of counting that ensured we counted everything while not double counting anything. Now, we are ready to go ahead with our study of derangements as promised. Understanding Derangements Let us first define what a…
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Recapitulation We are currently in the middle of a series on counting principles. At the conclusion of the previous post, I had promised you that we would look at the idea of derangements. However, I realized that we need to lay some ground work before we get to derangements. When we count, how do we ensure…
