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#1 Puzzles and Games » solution to finding the heavy or light ball from 12 balls. » 2005-05-23 11:01:01

geoff
Replies: 2

The numbers 1 to 12 can be made up from the numbers 1,3 and 9.
1 is made from 1
2 is made from 3 - 1
4 is made from 3 + 1
5 from 9 - 3 - 1
6 from 9 - 3 etc.
Using this the numbers 1 to 12 can be set up in a matrix as shown in the weighing sequence below.

First weighing      2,4,7,8   against 1,5,10,11    Using 1
Second weighing  3,4,6,11 against 2,5,7,12      Using 3
Third weighing     5,7,9,11 against 6,8,10,12    Using 9

The balls are numbered 1 to 12 and are weighed in fours on a two pan balance.
If ball 8 is heavy then the left pan will go down on the first weighing, they will balance on the second weighing and the right pan will go down on the third weighing. 
The value of 9 -1 gives you 8.  Try it with other values.

This method can be extended to include the number 27 so that any number up to and including 39 can be made up from 1,3,9,and 27.  You will get a result from 4 weighing.
Numbers up to 120 can be made from 1,3,9,27,and 81. this will require 5 weighings.

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