Polish Mathematicians Finding Patterns in Enigma Messages
In the 1930s, Polish codebreakers, the most significant of whom was Marian Rejewski, spotted patterns in German Enigma messages. Using elementary group theory, Rejewski was able to construct a model of Enigma and to determine the settings of the rotors.
Seeing Dots: Visibility of Lattice Points
Joshua Laison and Michelle Schick
Two lattice points in the plane are said to be mutually visible if the line segment between them contains no other lattice point, or equivalently, if the difference in their x-coordinates and the difference in their y-coordinates are relatively prime. Given an r by s rectangle R of points in the plane, how many additional lattice points are needed to view all points in R? Good upper and lower bounds are known for large rectangles, but these formulas can't be used to calculate specific values. We obtain results for some small values of r and s, and leave several open problems for the interested reader.
Faro Shuffles and the Chinese Remainder Theorem
For a deck of cards with an odd composite number mn of cards, where m and n are relatively prime, the Faro shuffle (or perfect riffle shuffle) works well with the Chinese Remainder Theorem. We describe this interaction and illustrate it with some card tricks.
The Lost Cousin of The Fundamental Theorem of Algebra
The fundamental theorem of algebra implies that a polynomial of order n has at most n roots. Such a polynomial is a linear combination of n + 1 power functions. An interesting question arises: does a sum of n + 1 functions of any other special type possess a similar property? We show that this is indeed the case for real exponential functions. On the other hand, for any integer n greater than one, we find a linear combination of n + 1 logarithm functions will have either exactly one or infinitely many roots depending on the coefficients, so we conclude that our result does not extend to, for example, the set of all monotone, continuous, or differentiable real functions.
Not Mixing is Just as Cool
Newton’s law of cooling, a staple of the calculus curriculum, is an empirical law not meant for mathematical proof. However we show it is mathematically equivalent to the intuitively appealing principle that the average temperature of two cooling objects is equal to the temperature of a single object with initial temperature the average of the other two.
Polynomial Congruences and Density
What do the solutions of a polynomial congruence look like, as the modulus varies? After normalizing so that the solutions of a polynomial congruence are represented by numbers in the unit interval, and setting aside some trivial counterexamples, we show that the solutions are dense in the interval.
A Curious Way to Test for Primes
Dennis P. Walsh
A cabdriver offers a curious route to identify prime numbers. Using a sequence of functions, each of which is a sum of exponential functions, we flesh out the cabbie's route. We show that an integer n is a prime if and only if the n-th derivative of the (n-1)st function of the sequence is 1 when evaluated at 0. The proof is surprisingly simple, using only a series expansion and differentiation. Similar to many cab rides, this route to the primes is roundabout but thoroughly interesting.
Excitement from an Error
Linda Marie Saliga
When an advanced calculus class discovered an error in a published note, the students expressed an extraordinary amount of excitement and passion for mathematics, and a class research project was born. This note is an account of the class’s discussions, which reveal a lot about the students’ thought processes, as the project progressed to its conclusion.
Shanille Practices More
Heather Anderton and Richard Jacobson
The following problem appears in the 63rd Putnam competition. Shanille O’Keal shoots free throws on a basketball court. She makes one out of the first two shots and thereafter the probability that she hits the next shot is equal to the proportion she has hit so far. What is the probability that she hits 50 of her next 100 shots? In this note we suppose that she hits q of her first p shots and find the probability that she hits k out of the next n shots.
Why Richard Cory Offed Himself or One Reason to Take a Course in Probability
J. D. Memory
This extends the well known poem "Richard Cory" by supposing that the suicide occurred because Cory fell victim to the False Positive Fallacy in interpreting the results of a test for a dread disease. Conditional probabilities are discussed in a footnote.
Caleb J. Emmons
A poem whose form illustrates the famous triangle of Blaise Pascal. Can any reader find the next line?