Before I explain some conventions in
setting out my solution to the Twelve Billiard Balls problem, here again
is a statement of the task.
There are twelve
billiard balls, all the same size, shape and colour. All weigh exactly
the same, except that one ball is slightly different in weight, but not
noticeably so in the hand. Moreover, the odd ball might be lighter or
heavier than the others.
Your challenge is to
discover the odd ball and whether it is lighter or heavier. You must use
a beam balance only, and you are restricted to three weighing
operations.
Conventions:
At every weighing one of three things
theoretically can happen: the pans can balance, the left pan can go down
or the left pan can go up.
It will be necessary to refer to a
given ball as definitely normal (N), potentially heavy (H) or
potentially light (L). Often our identification of a ball in this
way will be as part of a group (= This group contains a
heavy/light ball), and will depend on what we learn from a previous
weighing. At the start, all balls have a status of unknown (U).
To show at each weighing what is being
placed in each pan, we represent the situation as per the following
examples:

UUUU
UUUU

(This means four balls in each pan, all
of unknown status)


H
H

(This means two balls, one per pan,
each from a group temporarily identified as heavy)


UUU
NNN

(This means three balls of unknown
status weighed in the left pan against three balls whose status is known
definitely to be normal)

It is important also to be able to
imagine several separate areas on the bench where the balance is
standing. One is obviously for keeping balls that have already been
eliminated as normal; another is for balls that as a group are being
thought of as potentially heavy; likewise there is an area for
potentially light balls.
First
Weighing 
UUUU
UUUU 
Pans balance 
All these Us are now
known to be Ns; the odd ball is one of the remaining unweighed four
(call them UUUU from now on).
Proceed to Second Weighing:
Case 1 
Left pan down 
One of the four balls
in the left pan is heavy (call them HHHH from now on) or one of the four
balls in the right pan is light (call them LLLL
from now on).
Proceed to Second Weighing Case 2 
Left pan up 
One of the four balls in
the left pan is light (call them LLLL from now on) or one of the four
balls in the right pan is heavy (call them HHHH from now
on).
Proceed to Second Weighing Case 2 
Second Weighing 

Case 1 
UUU NNN 
Pans balance 
All these Us are now
known to be Ns; the odd ball is the remaining unweighed U, but we dont
yet know if its heavier or lighter than normal.
Proceed to Third Weighing Case 1 
Left pan down 
One of these Us is
heavier than normal, but we dont yet know which one (call them HHH from now
on).
Proceed to Third Weighing Case 2 
Left pan up 
One of these Us is
lighter than normal, but we dont yet know which one (call them LLL from now
on).
Proceed to Third Weighing Case 3 
Case 2 
HHL HLN 
Pans balance 
All these Hs and Ls
are now known to be Ns; the odd ball is one of the remaining unweighed
H or two Ls.
Proceed to Third Weighing Case 4 
Left pan down 
The odd ball is one of
the left two Hs or the right L.
Proceed to Third Weighing Case 5 
Left pan up 
The odd ball is either
the right H or the left L.
Proceed to Third Weighing Case 6 
Third Weighing 

Case 1 
U N 
Pans balance 
Not possible 
Left pan down 
The odd ball is this U,
and its heavier 
Left pan up 
The odd ball is this U,
and its lighter 
Case 2 
H H 
Pans balance 
The odd ball is the
remaining unweighed H (heavier) 
Left pan down 
The odd ball is the
left H (heavier) 
Left pan up 
The odd ball is the right
H (heavier) 
Case 3 
L L 
Pans balance 
The odd ball is the
remaining unweighed L (lighter) 
Left pan down 
The odd ball is the
right L (lighter) 
Left pan up 
The odd ball is the left L
(lighter) 
Case
4 
L L 
Pans balance 
The odd ball is the
remaining unweighed H (heavier) 
Left pan down 
The odd ball is the
right L (lighter) 
Left pan up 
The odd ball is the left
L (lighter) 
Case
5 
H H 
Pans balance 
The odd ball is the
remaining unweighed L (lighter) 
Left pan down 
The odd ball is the
left H (heavier) 
Left pan up 
The odd ball is the right
H (heavier) 
Case
6 
H N 
Pans balance 
The odd ball is the
remaining unweighed L (lighter) 
Left pan down 
The odd ball is this H
(heavier) 
Left pan up 
Not possible

