Rhind Mathematical Papyrus

In problem no. 32 of the Rhind Mathematical Papyrus, Ahmes divides 2 by 1 1/3 1/14 and obtains 1 1/6 1/12 1/114 1/228:

  2  divided by  1 '3 '4  equals  1 '6 '12 '114 '228

By following Ahmes, young pupils learn how to handle unit fractions and unit fraction series, whereas advanced learners may solve a more demanding problem. Let the edges of a right parallel-epiped measure

  height   2 units
  length   1 '3 '4 units
  breadth  1 '6 '12 '114 '228 units

How long are the cubic diagonals?

Simply

       1 '3 '4  plus  1 '6 '12 '114 '228  units

or

       1 1  plus  '3 '6  plus  '4 '12  plus  '114 '228  units

or

        2           '2           '3              '76    units

Divide 2 by any number a and you obtain b:

  2 : a = b

Using these numbers you can define a 'magic' parallelepiped of the following properties:

  height              2 units
  length or breadth   a units
  breadth or length   b units
  area base or top    ab square units
  volume              2ab cube units
  cubic diagonal      a+b units

Let the capacity of a granary measure 500 cube cubits and find solutions of the above type. All granaries have a height of 10 royal cubits, while length and width can vary. Here is one of many solutions:

  inner height   10 royal cubits or 70 palms
  inner length                      50 palms
  inner width                       49 palms
  cubic diagonal           exactly  99 palms

Allow a tiny mistake and you obtain this right-parallelepiped:

  (10 royal cubits = 70 palms = 280 fingers)

  rp  280 by 198 by 198 fingers

  
  cubic diagonal practically 396 fingers or 99 palms

This granary can easily be measured out using a simple rope with knots:

       10 royal cubits         198 fingers      198 fingers
  o------------------------o-----------------o-----------------o
       inner height           inner length      inner width
  o------------------------o-----------------------------------o
     diagonal base or top              cubic diagonal

(My interpretations of some 65 problems from the Rhind Mathematical Papyrus are found on my web site www.seshat.ch)

Franz Gnaedinger Zurich