Newsgroups: sci.math
From: eclrh@sun.leeds.ac.uk (Robert Hill)
Subject: Re: Speaking of wrong SAT answers...
Date: Tue, 10 Feb 1998 16:20:42 +0000 (GMT)

In article <6bourn$bg0$1@cantuc.canterbury.ac.nz>, 
mathwft@math.canterbury.ac.nz (Bill Taylor) writes:
> Maybe someone else remembers this one too.
> 
> Back in the early 70's I remember reading in "Time", of a Florida whizz-kid
> who sat the national SAT-equivalent of back then, and got not quite 100% in
> it. As he was convinced he'd got it all right, his father wrote off, and in
> due course they found a (moderately surprising) error in the standard answer.
[...]
> ---------------------------
> If you glue a regular tetrahedron to a square-based equilateral-sided
> pyramid, by exactly glueing 2 equilateral faces, how many faces does the
> resulting polyhedron have?
> ---------------------------
> 
> You're SUPPOSED to get 7, of course.
> 
> I wonder how many unforewarned sci.mathers would have got this wrong answer! :)

I would, for one.  I had not heard of this result till now.
 
> Interestingly, the fact that the resulting figure has less than 7, is one of
> those remarkable theorems that is *much* more easily proved "backwards"
> than directly.    Polya would have loved it!

With hindsight I would prove it as follows:

Lay two such pyramids side by side, in contact along a base edge,
with their square bases in one horizontal plane and their peaks upward.
Then all six edges of the tetrahedral gap between them are clearly equal,
so a regular tetrahedron would fill the gap snugly.  Each of its two
free faces fills the gap between two coplanar pyramid faces, and is
therefore coplanar with them.

This proof was discovered by observing first that if Bill's statement
is true, the stament about snug fitting would follow as a corollary.
Perhaps this is the "backwards" proof?
 
The idea also seems to lead to a nice filling of space by equal-edged
regular tetrahedra and octahedra.   

-- 
Robert Hill

University Computing Service, Leeds University, England

"Though all my wares be trash, the heart is true."
  - John Dowland, Fine Knacks for Ladies (1600)