Murray S. Klamkin

Extensions of the birthday surprise

Abstract The so-called “birthday surprise” is the fact that, on the average, one need only stop about 24 people at random to discover two who have the same birthday. Here we determine, asymptotically, the expected number of people in order for n of them to have the same birthday. In particular, for three birthdays, it is about 83 people.

A Characterization of Uniformly Accelerated Motion

Problem 95-14*, by RUDOLF X. MEYER (University of California at Los Angeles). Let S be a string of fixed length with ends A and B, with A fixed in the plane. Let S execute a continuous motion, such that S does not intersect itself nor the plane. Show that there exist for all integers n, periodic motions such that after the string is returned to its initial position, the twist of S, integrated from A to B, has vanished, yet the end B of S has rotated through 47rn. (The case n 0 is trivial.) Also show that there are no such motions for which the rotation is 2:rm, where m is an odd integer. The result has application to electrical cables that connect, without the use of sliding contacts, two bodies in relative, undirectional rotation to each other, such as occurs between a spacecraft and a rotating, scanning sensor. Note. The author notes that in one of his engineering design classes, he and his students designed a mechanical model that illustrates the theorem as shown in Fig. 1,

A Possible Characterization of Uniformly Accelerated Motion

Problem 94-9*, by R. W. Cox (Medical College of Wisconsin). Define the NxN symmetric nonnegative definite Toeplitz matrixA () by aij exp[-(ij)2] for e > 0. When e 0, the resulting matrix A(0) is rank and has only one nonzero eigenvalue. Call the eigenvalues k/,(), with k > X2 > > XN. Thus X(0) N and Xk (0) 0 for k > 1. For k 1, 2 N, and for positive e, show that the eigenvalues ofA() are asymptotically given by

Extreme Gravitational Attraction

Problem 92-2, by ANDY LIU (University of Alberta, Alberta, Canada). In Lotto 3-14, a player writes three distinct numbers from 1 to 14 inclusive on a ticket. In the subsequent drawing, three distinct numbers from 1 to 14 inclusive are drawn. The ticket wins a third prize if exactly one number matches, wins a second prize if exactly two numbers match, and a first prize if all three numbers match. Obviously,

A Duality Relation in Differential Equations and Some Associated Functional Equations

There are many instances of duality in differential equations. One well-known result is the Legendre transformation which is one of the simpler examples of a nontrivial contact (or tangent) transformation [1], [2]. For the general theory, one can refer to the fundamental work of Lie [1], [3]. In this paper, we extend a duality relation given by Shanks [4] for the simple exterior ballistic equation s = -Sn. We determine the general class of equations, s = F(s, s, t), for which the same duality holds. This leads to a functional equation which is solved by means of generalizations of the notions of even and odd. Shanks has shown that the ballistic equation