Using your Head is Permitted

May 2016 riddle

UPDATE (3 May): To clarify, when I write "places where we are used to using arithmetic/harmonic means", I mean more than just the positive reals. For example, you may want to consider also the example of S being the complex numbers in the first and fourth quadrants.

For anything other than the positive reals, I do not require, however, any proof of existence for the value you define.

In general, if F is a field and S is this field restricted in such a way that

  1. S is closed under reciprocation,
  2. The definitions given in the question coincide with standard definitions, and
  3. For any a and b there exists a unique element in S satisfying the standard definition of a geometric mean,
then your definition, if it defines a value in S, should coincide with the standard geometric mean on S.

Consider the following three definitions:

  • The arithmetic mean of a and b is (a+b)/2.
  • The harmonic mean of a and b is 2/(a-1+b-1).
  • The geometric mean of a and b is √ab.
These definitions can all be used when a and b are positive real numbers, but will they be useful more generally?

In mathematics, we often try to define things using the most restricted set of tools possible, in order to make them most generally applicable.

For example: let S be a metric space and let x-1 be an involution over S. We refer to the function x-1 as "the reciprocal" in S. (Explanation: a metric space is a set over which a function d is defined that provides a distance between any two elements, the distance being a non-negative real number. Here, a second function, x-1, is also defined on the set. This function maps elements of S back into elements of S in such a way that for any x in S, (x-1)-1=x. This is what being an "involution" means.)

I claim that this is enough to define both the arithmetic and harmonic means. As a first shot, we can try to redefine the arithmetic mean of a and b as "the unique element x in S for which d(a,x)=d(b,x) and d(a,x) is minimal". This will work if S is, for example, the reals, but over a general S nothing guarantees us that the element x defined in this way really is unique, or, in fact, exists at all. (Readers are welcome to come up with examples where it is not unique or does not exist.) We therefore refine the definition as follows:

"Let the arithmetic mean of a and b in S be the element x in S for which d(a,x)=d(b,x) and d(a,x) is minimal, if such an element exists and is unique."

and further define the harmonic mean by

"Let the harmonic mean of a and b in S be the reciprocal of the arithmetic mean of a-1 and b-1."

Things to note about this style of definition:

  • In places where we are used to using arithmetic/harmonic means, the new definitions coincide with the standard ones.
  • Elsewhere, the values may or may not be defined. (Admittedly, this property makes this month's riddle less well-defined than usual, too.)
  • The definitions make no use of any function not already introduced. Only the two functions guaranteed to exist, d and the reciprocal, are utilised. In particular, no division, addition or an explicit use of the constant "2" has been made.
  • These are "definitions", not "algorithms". The definition does not (necessarily) give an algorithm that would allow one to compute the means of a and b on any S, nor even to verify that a particular x is the means.
This month's riddle: using the same criteria as above, define the geometric mean on S.

List of solvers:

Yu Gao (1 May 18:45)
Radu-Alexandru Todor (2 May 05:51)
Dan Dima (2 May 18:45)
Michael Blaszczyk (2 May 20:26)
Claudio Baiocchi (3 May 17:54)
Joseph DeVincentis (4 May 04:52)
Lorenzo Gianferrari Pini (6 May 16:15)
Xiaoyi Cao (9 May 18:24)
Rui Viana (12 May 02:30)
Kujou Miu (12 May 17:50)
Mengxiao Zhang (18 May 11:35)
Zilin Jiang (19 May 12:39)
Itsik Horovitz (23 May 08:31)
Harald Bögeholz (29 May 22:09)
Oscar Volpatti (31 May 14:57)

Elegant and original solutions can be submitted to the puzzlemaster at Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.


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