I number them first. you first weigh 1234 and 5678.
There are two possibilities. they balance or they don't. If
they balance, then the good coin is in the group 9,10,11,12. So for
the second weighing you put 1,2 in the left and 9,10 on the
right. If these balance then the good coin is either 11 or 12.
Weigh coin 1 against 11. If they balance, the good coin is number 12.
If they do not balance, then 11 is the good coin.
If 1,2 vs 9,10 do not balance, then the good coin is either 9 or 10.
Again, weigh 1 against 9. If they balance, the good coin is number
10, otherwise it is number 9.
What if the first weighing 1,2,3,4 vs 5,6,7,8 does not balance? Then
any one of these coins could be that coin. to
proceed, you must keep track of which side is heavy for each of the
following weighings.
Suppose that 5,6,7,8 is the heavy side. We now weigh 1,5,6 against
2,7,8. If they balance, then the good coin is either 3 or 4.
Weigh 4 against 9, a known bad coin. If they balance then the good
coin is 3, otherwise it is 4.
Now, if 1,5,6 vs 2,7,8 does not balance, and 2,7,8 is the heavy side,
then either 7 or 8 is a good, heavy coin, or 1 is a good, light coin.
For the third weighing, weigh 7 against 8. Whichever side is heavy is
the good coin. If they balance, then 1 is the good coin. Should the
weighing of 1,5, 6 vs 2,7,8 show 1,5,6 to be the heavy side, then
either 5 or 6 is a good heavy coin or 2 is a light good coin. Weigh 5
against 6. The heavier one is the good coin. If they balance, then 2
is a good light coin
please a more easy one the next time...
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