Ask A Scientist Mathematics Archive

Author:      john q doe
I forgot the name of this particular problem, but lets call it the ball
problem.  The problem is you have 12 balls (of any kind), one ball either
lighter or heavier than the other 11.  A scale is provided to you but only
three weighs are allowed.  How can you possibly figure out which is the odd
ball?

Response #:  1 of 2
Author:      asmith
It is possible to do it if you know whether the odd ball is heavier or
lighter, but I do not know how to do it without that knowledge.  Here is a
scheme:  divide the 12 balls into 4 groups of 3, call them a, b, c, d. 


Compare the weights of a and b.  If they are different, compare a and c.  If
they are different, you know the odd ball's in a, and also you know whether
it is heavier or lighter, so compare two of the 3 in a and you will have the
answer in the third comparison.  If a and c are the same, the odd ball is in
b and again you can have the answer in the third comparison.  If a and b are
the same, compare a and c.  If they are different, the odd ball's in c, and
again you have it in the third weighing.  If a and c are the same however,
the odd ball must be in d, and without knowing whether it is heavier or
lighter you cannot get it in only one more weighing.

Response #:  2 of 2
Author:      rcwinther
I have lifted this practically verbatim from the book "Games for the Super-
intelligent" by the late Jim Fixx.  (I would never have figured this out.)
Number the balls 1 to 12.  Weigh 1, 2, 3, and 4 against 5, 6, 7, and 8.
If (1, 2, 3, 4) and (5, 6, 7, 8) balance:
  Weigh 9 and 10 against 11 and 8 (we know 8 is not the odd ball).  
  If (9, 10) and (11, 8) balance: then 12 is the odd one.
 
  Weigh 12 against any other to find out if it is heavy or light.

  If (9, 10) and (11, 8) do not balance: suppose 11 and 8 are heavier,
  than 9 and 10; then either 11 is heavy, or 9 is light, or 10 is light.

  Weigh 9 against 10; if they balance, 11 is heavy; if they do not,
  the lighter of 9 and 10 is the odd ball.

  (Similar argument if 11 and 8 are lighter than 9 and 10).

If (1, 2, 3, 4) and (5, 6, 7, 8) do not balance:
Suppose 5, 6, 7, and 8 are heavier than 1, 2, 3, & 4. Then:  one of 
(1, 2, 3, or 4) is light, or else one of (5, 6, 7, or 8) is heavy.
  Weigh 1, 2, and 5 against 3, 6, and 9.
  If they balance:  then either 7 is heavy, or 8 is heavy, or 4 is light.
     Weigh 7 against 8; if they balance, 4 is the odd ball, otherwise the
     heavier of 7 and 8 is the odd ball.

  If (1, 2, 5) and (3, 6, 9) do not balance: suppose 1, 2, and 5 are lighter
than 3, 6, and 9; then either 6 is heavy, or 1 is light, or 2 is light.
  Weigh 1 against 2 to find out which one of the three choices is true.
  Otherwise, suppose 1, 2, and 5 are heavier than 3, 6, and 9; then either 3
is light, or 5 is heavy.

  Weigh 3 against (say) 2 to find out which of the two choices is true.

  (Similar argument if 1, 2, and 5 are lighter than 3, 6, and 9).