Here is my solution I hope it is correct, if not please inform me why
Good morning people,
Here is the solution(s) to the 12 odd ball problem!
You could use chalk to identify ball potially lighter or heaver or both
Before I explain some conventions in setting out the solution to the Twelve Balls problem, here again is a statement of the task.
There are twelve balls, all the same size, shape and color. All weigh the same, except that one ball is minutely different in weight, but not noticeably so in the hand. Moreover, the odd ball might be lighter or heavier than the others.
Your challenge was 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.
Note: by starting with 6 /6 or, 5/ 5 or 3/ 3. or 2/2 or 1/1 it is impossible to solve the problem
“The only way to solve this rather complex problems is by starting by weighing 4 balls against 4 as you will see by the solution below”
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 “heavier” (H) or potentially “lighter” (L). Often our identification of a ball in this way will be as part of a group (= “This group contains a heavier/lighter 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, represent the situation as per the following examples:
.
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 might be heavier (call them HHHH from now on) or one of the four balls in the right pan might be lighter (call them LLLL from now on).
Proceed to Second Weighing — Case 2
Left pan up
One of the four balls in the left pan might be lighter (call them LLLL from now on) or one of the four balls in the right pan is heavier (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
Good morning people,
Here is the solution(s) to the 12 odd ball problem!
You could use chalk to identify ball potially lighter or heaver or both
Before I explain some conventions in setting out the solution to the Twelve Balls problem, here again is a statement of the task.
There are twelve balls, all the same size, shape and color. All weigh the same, except that one ball is minutely different in weight, but not noticeably so in the hand. Moreover, the odd ball might be lighter or heavier than the others.
Your challenge was 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.
Note: by starting with 6 /6 or, 5/ 5 or 3/ 3. or 2/2 or 1/1 it is impossible to solve the problem
“The only way to solve this rather complex problems is by starting by weighing 4 balls against 4 as you will see by the solution below”
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 “heavier” (H) or potentially “lighter” (L). Often our identification of a ball in this way will be as part of a group (= “This group contains a heavier/lighter 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, represent the situation as per the following examples:
.
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 might be heavier (call them HHHH from now on) or one of the four balls in the right pan might be lighter (call them LLLL from now on).
Proceed to Second Weighing — Case 2
Left pan up
One of the four balls in the left pan might be lighter (call them LLLL from now on) or one of the four balls in the right pan is heavier (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