Problem :
You have 12 identical-looking balls, one of them is counterfeited, it has a slightly different weight than the others (it could be heavier or lighter), using a pan balance, how can you determine the counterfeited ball using only 3 weighings ?
Note : think 🤔 by yourself, the solution is below 👇
Solution
Let’s define some notations using colors
Basic case :
Out of 3 balls if we know that the counterfeited ball is lighter or heavier, we can always find the counterfeited ball with only one weighting
Let’s assume that we have 3 balls one of them is counterfeited (lighter)
Case 1
(1) is lighter than (2)
We can then deduce that (1) is the counterfeited ball
Case 2
(1) is heavier than (2)
We can then deduce that (2) is the counterfeited ball
Case 3
(1) and (2) have the same weight
We can then deduce that (3) is the counterfeited ball
The 12 balls problem :
Let’s assume that we have 12 balls identical-looking, on of them is counterfeited (lighter or heaver)
We’ll split the balls in 3 groups
- Group [1] : (1,2,3,4)
- Group [2] : (5,6,7,8)
- Group [3] : (9,10,11,12)
Let’s weight the group [1] with [2]
Case 1
The group [1] and [2] have the same weight
We can then deduce that the balls of the group [1] & [2] are normal
Let’s weight normal balls (1,2,3) with the unchecked balls (9,10,11)
Case 1 -1
(1,2,3) and (9,10,11) have the same weight
We can deduce that the balls (1),(2),(3),(9),(10),(11) are normal
(12) is the counterfeited ball
We can use the last weighing to check whether the ball is lighter or heavier
Solved ✔️
Case 1–2
(1,2,3) is lighter than (9,10,11)
Basic case, Solved ✔️
Case 1–3
(1,2,3) is heavier than (9,10,11)
Basic case, Solved ✔️
Case 2
The group [1] is heavier than [2]
We can then deduce that the balls of the group [1] are supposed to be lighter and balls of the group [2] are supposed to be heavier
Let’s weight two supposed lighter balls & a supposed heavier ball (1,2,5) with the supposed lighter balls & a normal ball (3,4,9)
Case 2–1
(1,2,5) & (3,4,9) have the same weight
We can then deduce that those balls are normal and the counterfeited ball is among the supposed heavier balls (6,7,8)
Basic case, Solved ✔️
Case 2–2
(1,2,5) is lighter than (3,4,9)
We can then deduce that the counterfeited ball is (1) or (2)
We can use the last weighing to find the counterfeited ball
Solved ✔️
Case 2–3
(1,2,5) is heavier than (3,4,9)
We can then deduce that the counterfeited ball is either the lighter balls (3),(4) or the heavier ball (5)
We can use the last weighing between the ball (3) & (4) to deduce the counterfeited ball
Solved ✔️
Case 3
The group [1] is lighter than [2]
The exact same strategy of the case 2 can be used in this case to solve the problem
Solved ✔️
References
- 12 Balls puzzle (Youtube)
- Balance Puzzle (Wikipedia)
