The Flexible Band

Gustavus J. Simmons, in charge of research and development at Rolamite Inc., Albuquerque, N.M., sent this curious topological problem. Work at Rolamite involves complex banded rolling systems. One of the Rolamite engineers, Virgil Erbert, was confronted in the course of his work with the problem shown in Figure 105. End A of a flexible band was fastened to an object that was too large to pass through the slot at end B. It was essential that the band be formed into the looped"configuration shown in the illustration without detaching end A from the object to which it was fastened. Can it be done?

It looks impossible, but the answer is yes. The reader is invited to draw a rough facsimile of the band on a sheet of pa-

Figure 105

Figure 105

A looped-band topological puzzle

per, cut it out and tape end A to a tabletop. The puzzle, which is not difficult, is to manipulate the strip into the looped configuration.

Six playersâ€”call them A, B, C, D, E and Fâ€”sit around a circular table divided into six equal parts. At the center of the table is a disk mounted on a central pin around which it can rotate [see Figure 106]. The disk is marked with arrows and digits.

The wheel is spun five times. After each spin each player scores the number of points within his segment of the table. (If the wheel stops with its arrows exactly between adjacent players, the spin is not counted.) The players keep a running total of points, and the one with the largest total after the fifth spin is the winner. If there are ties for the highest score, no one wins and the game is played again.

The outcome of the first spin is shown in the illustration. C is ahead with five points. After the second spin D is ahead. After the fifth spin A is the winner. What was each player's final score? The information seems to be insufficient, yet the question can be answered accurately by deductive reasoning. This unusual logic problem is adapted from a puzzle in one of D. St. P. Barnard's popular "Brain-twister" columns in the British Observer.

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