What Color was the Bear?
Benjamin L. Schwartz · Mathematics Magazine · 1960
Our purpose in this note is two-fold. Firstly, we shall re-examine one of the oldest and best known chestnuts in mathematical puzzle literature. Itwill be discovered thatdespite its age and theconsiderablethought which has been devoted to it, the complete solution is generally not known. In fact, to the writer's knowledge, the results presented below have never been published before in their entirety. In carrying out this study, we shall employ some principles of mathematical investigation *which are highly respected and universally approved, but often ignored in practice! Our second purpose is to draw attention anew to these principles by pointing out, through the amusing example of the puzzle, the errors which can ensue from bypassing them. Our point of departure is the problem given below. Although several variants are current, we believe the formulation we give to be typical. The statement begins: An explorer on the surface of the earth (assumed spherical) sees a bear 100 yards due south of him. The bear then travels 100 yards due east while the explorer remains stationary. The explorer now fires a shot due south, which travels straight and true, and strikes and slays the bear. At this point, it is customary to ask, What color was the bear? The intended answer is, white, since the sequence of events is supposed to be possible only if the explorer were at the North Pole; hence, the bear must be one of the polar variety. Without considering whether the answer is correct, we can say that the reasoning in this explanation is definitely incorrect. The North Pole is not the only location at which the conditions can be met, as we shall show. Therefore, rather than the quaint color-of-the-bear formulation, we shall ask more prosaically for all possible locations on the earth where the described events could occur. It is intuitively obvious that any answer will be independent of longitude, hence all answers will be circles of latitude (in the case of the North Pole solution, a degenerate circle of zero radius). However, one of the points we wish to emphasize is that the use of intuition can be deceptive, and should be avoided. Hence, we shall allow the preceeding result to emerge as a result of our formal analysis rather than impose it apriori on the analytic formulation. Our method will be to employ the familiar spherical coordinate system with origin at the center of the earth. The coordinate 0 will be the longitude west of Greenwich (say), and b the colatitude south of the boreal pole. These concepts will be made more precise presently.