The March issue of The College Mathematics Journal features an abundance of student authors and several articles on algebraic topics. In that intersection, Arthur Benjamin and Ethan Brown (who was in middle school when they wrote the article) help prepare us for the upcoming Mathematics Awareness Month on "Mathematics, Magic, and Mystery." The cover art resonates with two articles: the quartet Field, Ivison, Reyher, and Warner help you determine the best direction to run from an oncoming truck, while trio Bolt, Meyer, and Visser consider how to run fast without ever breaking a four minute mile (six of those seven authors are students). Keep reading for topics ranging from permutations and matroids to golden triangles and cookies. —Brian Hopkins
Vol. 45, No. 2, pp. 82-159.
To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.
Fiacha Heneghan and T. Kyle Petersen
Calculus and combinatorics overlap, in that power series can be used to study combinatorially defined sequences. In this paper, we use exponential generating functions to study a curious refinement of the Euler numbers, which count the number of “up-down” permutations of length n.
Arthur T. Benjamin and Ethan J. Brown
We present several effective ways for a magician to create a 4-by-4 magic square where the total and some of the entries are prescribed by the audience.
Michael Bolt, Anthony Meyer, and Nicholas Visser
In this manuscript we present the mathematics that is needed to answer three counterintuitive problems related to the averaging of functions. The problems are manifestations of the question, “Is the average rate of change on a given interval determined by the average rate of change on subintervals of a fixed length?” We also ask questions in higher dimensions that may have interesting geometric significance.
This note uses a visual analysis of partition diagrams to give an elementary, pictorial proof of the classification theorem for nilpotent linear maps. We show that any nilpotent map is represented by a matrix with ones in certain positions on the first super-diagonal and zeroes elsewhere.
We provide a geometric proof of a classical result on proportions.
Elizabeth Field, Rachael Ivison, Amanda Reyher, and Steven Warner
If you are ever faced with an oncoming truck, this paper could save your life. We investigate the optimal path that you should take from the middle of the road to the curb in order to avoid being hit by an oncoming truck. Although your instincts may tell you to run directly toward the curb, it turns out that this path, although the shortest, is not generally the safest.
We give a visual proof of an infinite series involving the golden ratio.
Jeremy S. LeCrone and Nancy Ann Neudauer
By carefully defining independence, we create two structures on a finite group that satisfy the matroid axioms. Both of these matroids are transversal and graphic, they are duals of each other, and are fundamental transversal matroids. The matroids capture some of the group structure, but two isomorphic matroids may have come from non-isomorphic groups, so we may not be able to recapture the group from the matroid. Our definitions of independent sets cannot be extended in what seems the natural way based on independence of vectors. Finding a definition of independence that satisfies the matroid axioms may not always be possible, though there are always more possibilities on the horizon.
Leigh Marie Braswell and Tanya Khovanova
The Cookie Monster wants to empty a set of jars filled with various numbers of cookies. On each of his moves, he may choose any subset of jars and take the same number of cookies from each of those jars. The minimal number of moves to accomplish this depends on the initial distribution of cookies in the jars. We discuss bounds of these Cookie Monster numbers and explicitly find them for jars containing numbers of cookies in the Fibonacci, Tribonacci, and other nacci sequences.
Claudi Alsina, Roger Nelsen, and Hasan Unal
We prove wordlessly that the difference of consecutive integer cubes is congruent to 1 modulo 6.
A semigroup is a set with an associative binary operation (which may not contain an identity element). The commuting probability of a semigroup is the probability that two elements chosen at random commute with each other. In this paper, we construct a single family of semigroups which achieves each positive rational in the interval (0, 1] as a commuting probability.
Daniel McQuillan and Rob Poodiack
We prove the derivative formula for sine via a geometric argument and the symmetric derivative, and then use similar techniques for tangent and inverse tangent.
We strengthen a result on tangent lines of parabolas, originally proved with Taylor series, without making use of calculus.