12 Billiard Balls Puzzle

Twelfth Man Found Wanting — Tony Rogers

Copyright

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 U’s are now known to be N’s; 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 U’s are now known to be N’s; the odd ball is the remaining unweighed U, but we don’t yet know if it’s heavier or lighter than normal. Proceed to Third Weighing — Case 1
Left pan down	One of these U’s is heavier than normal, but we don’t yet know which one (call them HHH from now on). Proceed to Third Weighing — Case 2
Left pan up	One of these U’s is lighter than normal, but we don’t yet know which one (call them LLL from now on). Proceed to Third Weighing — Case 3
Case 2	HHL ——— HLN
Pans balance	All these H’s and L’s are now known to be N’s; the odd ball is one of the remaining unweighed H or two L’s. Proceed to Third Weighing — Case 4
Left pan down	The odd ball is one of the left two H’s 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 it’s heavier
Left pan up	The odd ball is this U, and it’s 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

The Twelve Billiard Balls Puzzle