Where the Camera Was
Katherine McL. Byers and James M. Henle
Given a photograph of a building, we derive a simple formula for the location of the camera. The formula requires two measurements of the building and five measurements from the photograph.
PROOF WITHOUT WORDS
Extrema of the Function a cos t + b sin t
M. Bayat, M. Hassani, and H. Teimoori
The formula for the distance from a point on the unit circle to the line ax + by = 0 is used to determine the maximum and minimum values of the function a cos t + b sin t .
Tic-Tac-Toe on a Finite Plane
Maureen T. Carroll and Steven T. Dougherty
We introduce a new version of tic-tac-toe, a game that has inspired many variations. We play on what is called a finite affine plane. Such a plane has n2 points that can be arranged on an n x n grid, but the plane has more lines than that grid would lead you to imagine. Counting these extra lines as wins makes the game more interesting. In standard 3 x 3 tic-tac-toe, a game between two skilled players always ends in a draw. What happens in this new game? The answer depends on the size of the plane. We show that the first player has a wining strategy on some planes, and the second player can force a draw on all the others. We also analyze play on projective planes, since they are a natural extension of affine planes. Finally, we show simple configurations of points that produce a draw with very few points.
Designs, Geometry, and a Golfer's Dilemma
Keith E. Mellinger
Arranging players to compete in various sports often poses some interesting and non-trivial mathematical problems. Here we look at arranging groups of golfers so that everybody gets a chance to play with everybody else in a systematic way. The problem posed here provides a connection to different combinatorial objects including designs, and finite affine and projective planes. Different solutions are presented along with their practical and mathematical implications.
Arithmetic Progressions with Three Parts in Prescribed Ratio and a Challenge of Fermat
Kenneth Fogarty and Cormac O’Sullivan
It is easy to see that 1+2 = 3 and 4+5+6 = 7+8. Can it ever happen that an arithmetic progression breaks into three consecutive parts, each with equal sums? We answer this question in the article, but not before having to follow in the footsteps of great number theorists of the past such as Pierre de Fermat and Leonhard Euler. We are led to the question of whether it is possible for four squares to be consecutive terms in an arithmetic progression—a challenge issued by Fermat to the other mathematicians of his day (and succeeding generations).
The Median Triangle in Hyperbolic Geometry
I. E. Leonard, J. E. Lewis, A. Liu, and G. Tokarsky
For a given triangle, the median triangle is a triangle constructed from the medians of the given triangle. The median triangle always exists in Euclidean geometry, and the existence can be shown by proving that the medians always satisfy the triangle inequality. The familiar high-school construction gives an alternate proof of its existence. It is not obvious that the median triangle always exists in hyperbolic geometry however. This note shows that it does indeed exist, but the note also shows that the familiar Euclidean geometry construction never yields the median triangle in hyperbolic geometry.
PROOF WITHOUT WORDS
The Sum of Cubes—An Extension of Archimedes' Sum of Squares
This Proof Without Words is of a proposition on a sum of cubes which has been generalized from Archimedes' sum of squares. As with the sum of squares, we state the proposition first as Archimedes would have, in geometric language, then give a proof befitting the geometric language.
A (Not So) Complex Solution to a2 + b2 = cn
Arnold M. Adelberg, Arthur T. Benjamin, and David I. Rudel
Using elementary properties of the Gaussian integers, all solutions to the Diophantine equation a2 + b2 = cn are succinctly described for all integer n ≥\geq 2.
On the Two-Box Paradox
Robert A. Agnew
The Two-Box Paradox refers to a game-show setting where a contestant selects one of two identical boxes containing monetary prizes, one twice the other. After observing the prize, the contestant may choose to trade boxes and the paradox arises because simple expected value considerations always motivate a trade. In this note, we analyze the trading decision from the perspective of expected utility of wealth. In that context, we show that the paradox only pertains to utility functions that are unbounded above and that, for common bounded utility functions, simple strategies are optimal wherein the contestant trades if and only if the observed prize is below a certain threshold.
A Conceptual Proof of Cramer’s Rule
We prove Cramer's rule for solving linear equation systems A x = b by observing that the quotient in Cramer's rule is invariant under row operations of the linear system. Hence Cramer's rule follows from the case when A is the identity matrix.
A Conceptual Proof of Cramer’s Rule
A Parent of Binet’s Formula?
There are many ways to derive the celebrated Binet formula, which expresses the Fibonacci numbers in terms of the golden ratio. We give a new way, by producing a polynomial identity that could perhaps be regarded as a parent of Binet's formula. More generally, for any Fibonacci-type, two-term linear recurrence, we derive the corresponding Binet formula from the same polynomial identity.
An Ideal Functional Equation with a Ring
What do all characteristic functions of the ideals of an integral domain have in common? They are precisely the solutions of a certain functional equation presented in the note. How about the characteristic functions of the subrings? Well, they are just the solutions of another functional equation. Characteristic functions of the prime ideals? Yet another functional equation.
Weiferich Primes and Period Lengths for the Expansions of Fractions
Gene Garza and Jeff Young
It is well known that some decimal expansions terminate, while others repeat in simple patterns and yet others repeat in complicated patterns. Here we expand upon what others have done while exploring the expansion of a fraction in any base. Our main result is a formula for the length of the repeating portion for expansions of reciprocals of composites in term of the lengths of the repeating portion for their prime factorizations.