# Using your Head is Permitted

## December 2011 solution

The answer is that in a tetrahedron of cost *x* one can fit tetrahedra
that cost anything less than 4/3 *x*. More generally, in *n*
dimensions the cost of the inside parcel may be anything less than
(*n*+1)^{2}/4*n* *x* if
*n* is odd, and (*n*+2)/4 *x* if *n* is even.
To see that this is possible, consider a tetrahedron *ABCD* inside a
tetrahedron *A*'*B*'*C*'*D*', where all these points are
very close to being on a straight line and, furthermore, where
*A*,*B*,*A*',*B*' and *C*' are all very close together
and so are *C*,*D* and *D*' (though they are far from the first
set). If the distance between the two sets of points is approximately *a*,
then the sum of the sides of *ABCD* is approximately 4*a*, while the
sum of the sides of *A*'*B*'*C*'*D*' is approximately
3*a*.

To prove that this is also the upper bound, consider the sum of the lengths of
the sides of these tetrahedra, when projected onto an arbitrary line. In the
single-dimensional tetrahedron case, it is easy to show that the inside
tetrahedron never has more than 4/3 the side-sum of the outside tetrahedron.
Because it is true in every case, it is also true for the expected case, when
choosing a line in a random, uniformly chosen direction (over the uniform
measure on the sphere). This expectation, however, is proportional to the
side-sum of the three-dimensional tetrahedron (a fact that can be ascertained
easily by seeing that for a single straight line, this statistic is invariant
to rotations or translations, from which it follows also for the sum of several
straight lines).

Q.E.D.

The extension to higher dimensions is trivial. Readers may also contemplate what
would happen if the *n*-dimensional post-office were to charge by the sum
of the *k*-dimensional surfaces of the simplex.

As promised, some references:

This riddle has previously appeared in
this
French-language riddle site.
Its solutions direct to a paper in Mathematics magazine, Vol 73, No. 3,
June 2000. This, in turn, refers to "Macalester Problem of the week", No. 834,
as well as to "Crux Mathematicorum", December 1996. Further attributions can
be found there
(Problem 2099).

I further thank Claudio Baiocchi for undertaking this extensive literature
review, to make sure all credit goes to where it is due.

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