Newsgroups: rec.puzzles, sci.math
Followup-To: rec.puzzles
From: hoey@zogwarg.etl.army.mil (Dan Hoey)
Date: 24 Jul 91 14:00:25 GMT
Subject: Re: A Puzzling Catalyst (Spoiler, but the puzzle continues)

cos...@bbn.com (Bernie Cosell) writes:

>dcasa...@magnus.acs.ohio-state.edu (Donald J Casadonte) writes:

>}The will al[l]otted specific portions of the cattle the sheik had
>}owned to go to each son.  To the oldest, 1/2 of the cattle were to
>}be given.  To the next oldest, 1/3 were to be given, and to the
>}youngest, 1/9 were to be given.
>}"Look, suppose I give you one of my cattle to put in with your
>}own...  [divide them up] and then I will take my cattle back..."

>What is happening, of course, is that the will ... only allocated
>17/18ths of the flock.  By having the denominator have interesting
>factors you can hide the omission pretty well.  So let's pick another
>good-multiplier at random: say 24.  So we need to add up to 23/24ths,
>and so we can do it with 12, 8, 3.
...
> his flock was five, and he bequeathed 1/2 to one son and 1/3 to the
> other....

Or even, suppose he bequeathed half of his only cow to his only son!
I wonder if we should count the case where he had neither cattle nor
sons?

>his flock was 23 and to his five sons he bequeathed 1/3, 1/4, 1/6,
>1/8 and 1/12 of his flock.

The unstated criterion here is that the will uses Egyptian fractions
(the numerators are all one).  So the question here is for which
numbers N are there factors of N adding to N-1?  It certainly works
for any power of 2, by taking all the proper factors.  In fact 2^K is
the smallest K-son number.  Bernie noted N=6; this is the largest
two-son number.  What is the largest three-son number? (In case you
need a hint, V pbhyq gryy lbh ohg gung jbhyq tvir gur nafjre njnl.)
Note that twelve is a three-son number in two different ways: a sheik
with eleven cattle could bequeath 1/2, 1/4, and 1/6 or 1/2, 1/3, and
1/12.  What is the smallest three-way number?  What is the smallest
K-way K-son number (and what is the smallest K?  They need not be the
same solution, as far as I know.  But then I'm not sure any exist).

The big question is, is there an ODD K-son number (other than one)?
I don't expect anyone knows, but this is not quite a famous unsolved
problem.  Is there any literature on this one?

Dan Hoey
Hoey@ETL.Army.Mil