Site hosted by Angelfire.com: Build your free website today!
The famous 12 balls 3 weighing Problem..
The problem
There are 12 identically looking balls and 1 of them is lighter or heavier than the rest. You have a scale (no weights) and in 3 weighings should find out which ball is bad and if it is heavy or light!

The solution
First number the balls 1,2,3 etc. and each can be L(light) or heavy(H).you need a paper to write down the inferences of each weighing-can't solve mentally.

weighing 1
1234 ***  5678 - if left pan is up then 1L or 2L or 3L or 4L or 5H or 6H or 7H or 8H
(remember the inference from this first step-it is used below)!
(weighings 2 and 3 are given below depending on what happens in second weighing.)
weighing 2
127 *** 549  - if equal,one of 3,6,8 is bad.
weighing 3
36 *** 45   -if equal then 8H is the solution
36 *** 45  - if left pan up then 3L or 6L - 6 cannot be both H and L so 3L is the answer
36 *** 45  -if right pan up,then 3H or 6H- 6H is the answer as 3 cannot  be both H and L
 		 weighing 2
127*** 549-if left pan is up,
then 1L or 2L or 7L or 5H or 4H (9 is good)
weighing 3
1,5 *** 4,3 - if equal then 2 or 7 is bad-2L is the answer as 7 cannot be both H and L
1,5 *** 4,3 -if left pan up,1L or 5L -1L is answer
1,5 *** 4,3-if right pan up,1H or 5H and so 5H is answer as 1 cannot be both H and L
 		 weighing 2
127 ***549-if left pan down,
1H or 2H or 7H or 5L or 4L . hence 7H or 4L is the answer as 1,2 and 5 cant be both H and L
weighing 3
4 *** 10 -if left pan up,then 4L is answer
4 *** 10-if equal,then 7H is the answer
(left pan cannot be down as we have established 4 is Light and 10 is good.)
 

Note that if in weighing 1,5,6,7,8 side were up,the same procedure as above may be followed.

Now what happens if the pans in first weighing come up equal?
weighing 1
Assume 1234 **** 5678 pans are equal. Then one of 9,10,11,12 is bad and we don't know if each is L or H.
 
weighing 2
9,10 *** 1,11 -if equal ,then 12 is bad
weighing 3
12 *** 10 -if 12 is H or L can be found in this weiging as 10 is good		 weighing 2
9,10 *** 1,11 - if left pan up, then 9L,10L or 11H
weighing 3
9,11 *** 1,2 - if equal, 10L is answer
9,11 *** 1,2 -if right pan up,9H or 11H. 9 cannot be both H and L. so 11H is the answer

9,11 *** 1,2-if right pan down,9L or 11L. 9L is the answer
 

 weighing 2
9,10 *** 1,11-if left pan down,then 9H or 10H or 11L
weighing 3
9,11 *** 1,2-if left pan down,9H,11H. 11 cant be both L and H so 9H is answer
9,11***1,2-if right pan down,9L or 11L. 11L is answer
(note the pans cant be equal as one of them is bad)