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APPENDIX 5

Series and the Weights Problems

A mathematical problem of great antiquity is that wherein
it is required to find the least number of different weights
which would enable us to weigh any integral number of
pounds, from 1 pound to a specified upper limit. The usual
problem has an upper limit of 40 1b.

It is first to be noted that there are two different ways of
using the weights and the problems usually stipulate either
(@) that the weights may be placed in only one of the scale-
pans, or () that they may be placed in either of the pans
so that at any one time there may be one or more weights
in one or both scale-pans. There are therefore really two
problems, but both solutions are in the form of geometric
progressions.

If only one pan may contain weights then, as stated by
Tartaglia in 1556, the weights required are:

R SR i S al ue

whereas if both pans may be used simultaneously, the
weights required are:
B9

By the second method the weight of 2 Ib., for instance, is
obtained by placing the 1-lb. and g-lb. weights in opposite
pans, and the same principle is repeated where necessary
for other weights. This solution is due to Bachet, who pub-
lished it in 1624.

Bachet could not prove that his solution was the only one
possible with all-different weights or that it used the least
number of weights, but this was subsequently proved in 1886
by Major MacMahon.

172

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