120 PROBLEMS AND SOLUTIONS

On Winning in the Game of Lotto

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 aredrawn. The ticket wins a third prize if exactly one number matches, wins a second prizeif exactly two numbers match, and a first prize if all three numbers match. Obviously,

to guarantee winning a first prize, one must buy a ticket for each of the (134) 364combinations.

(a) What is the minimum number of tickets that will guarantee winning at least athird prize?

(b) Prove that 10 tickets are not enough to guarantee winning at least a second prize.(c) Show how to guarantee winning at least a second prize with 14 tickets.(d)* What is the minimum number of tickets that will guarantee winning at least a

second prize?

Expected Type of Triangle

Problem 92-3*, by D. J. NEWMAN (Temple University, Philadelphia, PA).Usually when one draws a triangle, the triangle is drawn to be acute. The same

is true in most geometry books. Show that for "most continuous distributions," a ran-dom triangle is more probable to be obtuse. For example, if three points are chosenindependently and uniformly within or on a given sphere, the probability of an obtusetriangle being formed is 37/70; if the sphere is replaced by a circle the probability is9/8 4/7r2 0.7197.

A Definite Integral Arising in Ohmic Dissipation

Problem 92-4, by A. A. JAGERS AND E. M. J. NIESSEN (Universiteit Twente, Enschede,the Netherlands).

Prove that

In2 (1 + 2p cos x + p2)dx 27r p2,/n2r n=l

for 0 < p < 1. What about the case where p > 1?The integral arose in calculating the stationary ohmic dissipation in a circular unsat-

urated superconducting wire.

Extreme Gravitational Attraction

Problem 92-5*, by M. S. KLAMKIN (University of Alberta).It follows by symmetry that the inverse square law attraction of any uniform homo-

geneous regular polyhedron on a unit test particle located at its centroid is zero.(i) Determine the location of the test particle, within or on a regular polyhedron, in

particular the cube, which maximizes the attraction.(ii) Consider the same problem (and also the minimum attraction) for a uniform

homogeneous torus.

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