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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?


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

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

        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
       inner height           inner length      inner width
     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

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