It was originally a billiard ball puzzle... and you also had to determine if the odd ball (penny) was heavier or lighter.
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 âheavyâ (H) or potentially âlightâ (L). Often our identification of a ball in this way will be as part of a group (= âThis group contains a heavy/light ballâ), 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 âââ UUUU
(This means four balls in each pan, all of unknown status)
H âââ H
(This means two balls, one per pan, each from a group temporarily identified as âheavyâ)
UUU âââ 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 âââ UUUU
Pans balance All these Uâs are now known to be Nâs; 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 â 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 â Case 2
SECOND WEIGHING
Case 1 UUU âââ NNN
Pans balance All these Uâs are now known to be Nâs; the odd ball is the remaining unweighed U, but we donât yet know if itâs heavier or lighter than normal. Proceed to Third Weighing â Case 1
Left pan down One of these Uâs is heavier than normal, but we donât yet know which one (call them HHH from now on). Proceed to Third Weighing â Case 2
Left pan up One of these Uâs is lighter than normal, but we donât yet know which one (call them LLL from now on). Proceed to Third Weighing â Case 3
Case 2 HHL âââ HLN
Pans balance All these Hâs and Lâs are now known to be Nâs; the odd ball is one of the remaining unweighed H or two Lâs. Proceed to Third Weighing â Case 4
Left pan down The odd ball is one of the left two Hâs or the right L. Proceed to Third Weighing â Case 5
Left pan up The odd ball is either the right H or the left L. Proceed to Third Weighing â Case 6
THIRD WEIGHING
Case 1 U âââ N
Pans balance Not possible
Left pan down The odd ball is this U, and itâs heavier
Left pan up The odd ball is this U, and itâs lighter
Case 2 H âââ 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 âââ 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 âââ 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 âââ 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 âââ 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
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 âheavyâ (H) or potentially âlightâ (L). Often our identification of a ball in this way will be as part of a group (= âThis group contains a heavy/light ballâ), 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 âââ UUUU
(This means four balls in each pan, all of unknown status)
H âââ H
(This means two balls, one per pan, each from a group temporarily identified as âheavyâ)
UUU âââ 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 âââ UUUU
Pans balance All these Uâs are now known to be Nâs; 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 â 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 â Case 2
SECOND WEIGHING
Case 1 UUU âââ NNN
Pans balance All these Uâs are now known to be Nâs; the odd ball is the remaining unweighed U, but we donât yet know if itâs heavier or lighter than normal. Proceed to Third Weighing â Case 1
Left pan down One of these Uâs is heavier than normal, but we donât yet know which one (call them HHH from now on). Proceed to Third Weighing â Case 2
Left pan up One of these Uâs is lighter than normal, but we donât yet know which one (call them LLL from now on). Proceed to Third Weighing â Case 3
Case 2 HHL âââ HLN
Pans balance All these Hâs and Lâs are now known to be Nâs; the odd ball is one of the remaining unweighed H or two Lâs. Proceed to Third Weighing â Case 4
Left pan down The odd ball is one of the left two Hâs or the right L. Proceed to Third Weighing â Case 5
Left pan up The odd ball is either the right H or the left L. Proceed to Third Weighing â Case 6
THIRD WEIGHING
Case 1 U âââ N
Pans balance Not possible
Left pan down The odd ball is this U, and itâs heavier
Left pan up The odd ball is this U, and itâs lighter
Case 2 H âââ 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 âââ 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 âââ 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 âââ 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 âââ 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