This class of problem derives from ancient Egyptian
methods to exactly partition estates, whenever possible.
Solving the relatively easy conversion of 17/18 and
39/40 by tables, into concise Egyptian fraction series,
only scratches the surface of the history of mathematics
and logical points that Ian Stewart discussed.
One solution to Menninger's 39/40 camel problem reveals
terms from the Old Kingdoms Horus-Eye duplation based series
(1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...). The HM
suggested solution to 17/18 'mule problem' was cited
by Stewart (as I recall) as a camel problem.
Generally, the EMLR (The Egyptian Mathematical Leather
Roll), and its 26 1/p and 1/pq series and the RMP (Rhind
Mathematical Papyrus), and its 51 2/p and 2/pq series
provides an Egyptian number theory foundation to this
problem, as algorithmic methods to create needed tables
of n/p and n/pq series. In addition, the 300 BC Hibeh P.
and its n/45 table provides a Greek science application
of this class of Egyptian fraction series. Finally, a
Hellenized Coptic Akhmim P. from around 500 AD provides
several n/p and n/pq tables converted into concise Egyptian
fraction series, that provides strong hints to the actual
algorithmic methods that were in use for over 3,000 years.
Were all of these exact tables used to find solutions to
estate partition problems, as recreation? I suggest that
this class of generalized Egyptian fraction table reveals
major components of a numeration system, one that provided
exact numerical solutions to science problems, such as
astronomy calculations listed in the Hibeh P. n/45 table
(as easily read), as well as everyday problems of Egyptian
(and maybe Greek) laws of inheritances.
Regards to all,
Milo Gardner
Sacramento, Calif.