Three Pyramids are Better Than Two

The applet presents a tool for investigating an infamous problem. It showed up in 1980 as a practice problem on the Preliminary Scholastic Aptitude Exam, but, to an embarrassment of the Educational Testing Service, the marked answer was incorrect.

The problem is this [Winkler, p. 43]:

(In the applet, objects can be rotated with the left button pressed or translated with the right button pressed. Pressing the left button while the Alt key is down lets one zoom in and away from an object. When the "combine" button is checked the pyramids will move as a single object.)

The expected solution was this. A square pyramid has 5 faces and a tetrahedron 4 faces. When two triangular faces are eliminated by gluing them together there remain 5 + 4 - 2 = 7 faces. By adding a second square-base pyramid, it becomes absolutely clear that, in the problem, there are two pairs of adjacent faces that align on a single plane (i.e., are coplanar) so that the resulting solid has two faces less than expected: 5 in all.

(As an aside, what is the relationship between the volumes of the square pyramid and the tertrahedron? Find out.

References

1. P. Winkler, Mathematical Puzzles: A Connoisseur's Collection, A K Peters, 2004

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