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Ternary Balance Puzzle
Posted:
Jan 20, 2013 5:31 PM
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Main reference:
http://en.wikipedia.org/wiki/Balanced_ternary
<quote> Balanced ternary has other applications besides computing.
For example, a classical two-pan balance, with one weight for each
power of 3, can weigh relatively heavy objects accurately with a
small number of weights, by moving weights between the two pans and
the table. For example, with weights for each power of 3 through 81,
a 60-gram object (60 = 1T1T0) will be balanced perfectly with an 81
gram weight in the other pan, the 27 gram weight in its own pan,
the 9 gram weight in the other pan, the 3 gram weight in its own pan,
and the 1 gram weight set aside. </quote>
http://en.wikipedia.org/wiki/Weighing_scale#Balance
Any whole mass (positive integer) between 0 and 13 can be measured
with just 3 weights in the left and/or right pan of the balance:
13 = 1 + 3 + 9 ; 12 = 9 + 3 ; 11 = 9 + 3 - 1 ; 10 = 9 + 1 ; 9 = 9 ;
8 = 9 - 1 ; 7 = 9 - 2 ; 6 = 9 - 3 ; 5 = 9 - 3 - 1 ; 4 = 3 + 1 ;
3 = 3 ; 2 = 3 - 2 ; 1 = 1 .
"Bachet's weights problem" asks for the minimum number of weights
(which can be placed in either pan of a two-arm balance) required
to weigh any integral number of pounds from 1 to 40. The solution
is 1, 3, 9, and 27.
But now comes the real question. Yes, it's easy to "measure" a well
known integer with a ternary balance. But what if the mass at hand
is integer and .. UNKNOWN ? I could not find a general strategy for
efficiently placing the weights in the left pan, the right pan, or
on the table, for unknown weights. If there does not exist sort of
such an optimal strategy, then the whole idea of a ternary balance
becomes questionable in reality.
Any takers?
Han de Bruijn
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