Question

Given 12 balls with the same diameter and color. They all have the same weight, except one of them which can be lighter or heavier. Show how to identify the ball with different weight with a balance. Any number of balls can be placed on both sides of the balance. Your goal is to design a procedure to identify the ball with different weight in 3 measurements. Can you do it with 2 measurements? Justify your answer.

Answer 1

For our easy understanding we name balls as 1-12

=>if both side are balanaced:first wegiht [1,2,3,4] on the left side and [ 5,6,7,8] on the right side

1)if the they are balanaced we can say 9,10,11,12 have fault ball,now weight [8,9] and [10,11] you already knwo that 8 is not falut ball,now if they balanace 12 will be the fault ball

2)if [8,9] is heavier the either 9 is heavy or 10 will the light or 11 will be ,now wieght 10 and 11 if they balanace 9 is fault (heavier) if they dont balanace one is falut (lighter),

3)if [8,9] is lighter then either 9 0r 10 is heavy or 11 is heavy . now wieght 10 and 11 if they balanace 9 is fault (lighter) if they dont balanace one is falut (heavier)

if [1,2,3,4] is heavier than the other side:,

we already know that either one of [1,2,3,4] heavier or one of [5,6,7,8] is lighter but it will confirms that [9,10,11,12] are not fault. This is main point now,Weight [1,2,5] and [3,6,9] (Note: 9 is surely not fault).

=>if they balance, then either 4 is heavy or 7 is light or 8 is light. Please follow the last step from the previous case, we weigh [7] and [8]. if they balance, 4 is falut(heavier). if they don’t balance then whichever one is lighter is falut (lighter).

=>if [1,2,5] is heavier, then either 1 is heavy or 2 is heavy or 6 is light. Weigh [1] and [2]. if they balance, 6 is falut (lighter). if they don’t balance then whichever one is heavier is falut (heavier).

=>if [3,6,9] is heavier, then either 3 is heavy or 5 is light. Weigh [5] and [9]. They won’t balance. if [5] is lighter, 5 is falut (lighter). if they balance, 3 is falut (heavier).

if [5,6,7,8] is heavier than the other :

it is the similar situation as if [1,2,3,4] was heavier. Just perform the same steps using 5,6,7 and 8. Weigh [5,6,1] and [7,2,9] (Note: 9 is surely not falut).

=>if they balance, then either 8 is heavy or 3 is light or 4 is light. Following the last step from the previous case, we weigh [3] and [4]. if they balance, 8 is falut(heavier). if they don’t balance then whichever one is lighter is falut (lighter).

=>if [5,6,1] is heavier, then either 5 is heavy or 6 is heavy or 2 is light. Weigh [5] and [6]. if they balance, 2 is falut (lighter). if they don’t balance then whichever one is heavier is falut (heavier).

=>if [7,2,9] is heavier, then either 7 is heavy or 1 is light. Weigh [1] and [9]. if they balance, 7 is falut (heavier). if they don’t balance then 1 is falut (lighter).