A curve can be used as a visually-strong way of specifying a point – for example, as the center of a circle. This distinction is useful, for example, in discussing the ‘9-point’ puzzle – where the points are presented to the audience as ‘dots’ (“connect the dots…”) – are the ‘dots’ literally dots? If so, 3 lines suffice, but if they are only metaphorical dots, then that famous 3-line solution does not work.

keywords: Mathematics, Logic, Geometry, Linguistics

# Day: July 19, 2019

# riffing off the Collatz process

Definition 1. Let X be an n-tuple of positive integers. Then the k-th term of X is said to be a triangle-admissible member of X fif the terms of X up to and including the k-th term can be partitioned into three sets A, B, and C such that sumA, sumB, and sumC satisfy the triangle inequality.

Definition 2. A given positive integer is said to be Collatz-compatible fif its associated Collatz process reaches 1 (i.e., if its total stopping time is finite).

Definition 3. For each Collatz-compatible positive integer n, let X be the m-tuple of the terms of the Collatz process for n, with m being its total stopping time, and let p be the number of the terms of X that are triangle-admissible. Then f(n) = p/m.

QUERY: Given the sequence {n} of Collatz-compatible positive integers, what does the sequence {f(n)} look like?

QUERY 2. Among the triangle-admissible terms, are there any that admit a Pythagorean triple?

keywords: Mathematics, open problem

(posted to CI on 19.Jul.2019)

# proving that f(x) > g(y)

We are given sets A, B, C, and D, with x in A and y in C, and asked to prove that f(x) > g(y).

It is sufficient if to be able to do the following:

1. prove that x is in A implies f(x) is in B;

2. prove that if y is in C, then g(y) is in D;

3. prove that B > D (that is, every element of B is greater than every element of D).

keywords: Mathematics, Logic, proof technique

# necessary versus sufficient conditions

Everyone knows the difference between a necessary condition and a sufficient condition, but in the heat of battle the distinction between them can be forgotten. The distinction can obtain also at the meta level. A nice example is being asked to ascertain whether the cotangent of 40 degrees is greater than the cosine of 5 degrees. It would be sufficient is we knew the two values, because we could then just inspect them and draw the appropriate conclusion, but it is not necessary. It is sufficient merely to note that the cotangent of 40 degrees must be greater than 1, and that the cosine of 5 degrees must be less than 1 to conclude that, indeed, the cotangent of 40 degrees is greater than the cosine of 5 degrees.

keywords: Mathematics, Logic, Trigonometry, trigonometric functions

# the five-number summary in Probstat

Here is the link to a definition of it.

keywords: minimum, maximum, Q1, Q2, Q3, median

# great colleges for top students

There is a book dedicated to this topic:

PETERSON’S 440 GREAT COLLEGES FOR TOP STUDENTS

Here is the link to a preview of this book.

keywords: education, college admissions, freshman class, competition, excellence

# wanted software feature

a five-number summary of the news.

– similar to the ‘five-number summary’ in Probstat.

This would allow having an idea of the ‘weather’ in the social sphere, without having to actually endure the distraction and time-sink of watching the news.

cf: (“We must tend our garden.” – Voltaire)

keywords: meta news, news analytics