Math Is Fun Forum

A friend of mine told me this problem, and I worked out a solution. Then I looked up the solution on this website. I like my solution much better!

The Problem:

You have 12 balls identical in size and appearance but 1 is an odd weight (could be either light or heavy).

You have a set scales (balance) which will give 3 possible readings: Left = Right, Left > Right or Left < Right (ie Left and Right have equal weight, Left is Heavier, or Left is Lighter).

You have only 3 chances to weigh the balls in any combination using the scales. Determine which ball is the odd one and if it's heavier or lighter than the rest. How do you do it?

My Solution:

Label the balls 1-12

First Weighing:
Left:     1 2 3 4
Right:    5 6 7 8
Off:    9 10 11 12
Record the heavier side (L, R, or B)

Second Weighing:
    Left:    1 2 5 9
    Right:    3 4 10 11
    Off:    6 7 8 12
    Record the heavier side (L, R or B)

Third Weighing:
    Left:    3 7 9 10
    Right:    1 4 6 12
    Off:    2 5 8 11
    Record the heavier side (L, R, B)

There are 27 (3^3) possible combination of scale readings. A complete sorted list of the scale reading appears below. Note that only 24 of the 27 readings should be possible given the original problem statement. The algorithm was designed so that if all three scale readings are the same, an error is flagged indicating that the scale is stuck.
   
    BBB    Error! There is not a single light or heavy ball (or scale is stuck).
    BBL    Ball #12 is light
    BBR    Ball #12 is heavy
    BLB    Ball #11 is light
    BLL    Ball #9 is heavy
    BLR    Ball #10 is light
    BRB    Ball #11 is heavy
    BRL    Ball #10 is heavy
    BRR    Ball #9 is light
    LBB    Ball #8 is light
    LBL    Ball #6 is light
    LBR    Ball #7 is light
    LLL    Error! Scale is stuck!
    LLB    Ball #2 is heavy
    LLR    Ball #1 is heavy
    LRB    Ball #5 is light
    LRL    Ball #3 is heavy
    LRR    Ball #4 is heavy
    RBB    Ball #8 is heavy
    RBL    Ball #7 is heavy
    RBR    Ball #6 is heavy
    RLB    Ball #5 is heavy
    RLL    Ball #4 is light
    RLR    Ball #3 is light
    RRB    Ball #2 is light
    RRL    Ball #1 is light
    RRR    Error! Scale is stuck!