Amanda Montañez
Imagine a large cube formed by gluing together 27 smaller wooden cubes, all of uniform size. A termite starts at the center of the face of any one of the edge cubes and bores a path that takes it through every cube only once. Its movement is always parallel to a side of the large cube, never diagonal. Is it possible for the termite to bore through each of the 26 outside cubes once and only once, then finish its trip by entering the innermost cube for the first time? If possible, show how this can be done; if impossible, prove it.
It is not possible for the termite to pass through the 26 outside cubes only once and end its journey in the innermost one. This is easily demonstrated by imagining that the cubes alternate in color like the cells of a three-dimensional checkerboard. The large cube will then consist of 13 cubes of one color and 14 of the other color. The termite’s path is always through cubes that alternate in color along the way; therefore, if the path passes through all 27 cubes only one time, it must begin and end with a cube belonging to the set of 14. The central cube, however, belongs to the set of 13, making the desired path is impossible.
We’d love to hear from you! E-mail us at games@sciam.com to share your experience.
A version of this puzzle originally appeared in the October 1960 issue of Scientific American.