10 identical bottles of pills Puzzle - Interview Puzzles

Puzzle Details 

Given 10 identical bottles of identical pills (each bottle contains hundred of pills). Out of 10 bottles, 9 have 1 gram of pills but 1 bottle has pills of the weight of 1.1 gram. Given a measurement scale, how would you find the heavy bottle? You can use the scale only once.

Note: questions can also be asked like, what the minimum number of weighs require to find the bottle with the pills which are heavier.

Solution

initially, this looks like a complicated puzzle, but this is a simple one if you know the logic.

since only 10 bottles are given, the puzzle becomes more simple. If you start weighing every bottle, then this is not the logical approach to get the correct answer. let me explain the most systematic way to get the answer.

let us go step by step.

Step 1: arrange the bottles on the shelf and now take, 1 pill from the first bottle, 2 pills from the second bottle, 3 pills from the third bottle, and so on
Till 10 pills from the 10th bottle. then add the number of pills.

it looks like this ..... 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

this is the sum of n natural numbers, in this case, n is 10
we know that sum of n natural numbers can be calculated using the formula n(n+1)/2, after solving the equation we get 55. you can count instead of using the formula but if the number of bottles is more then it is better to go with the formula to find the sum of n natural numbers.

Step 2: if we weigh these pills,

Ideally, you would have (10)*(11)/2=55 pills, weighing 55 grams. if we assume all pills weigh 1 gm but there are pills from one bottle which weigh 0.1 gm more. When you put the entire pile of pills on the weighing scale, the deviation from 55g would tell you which bottle contains the heavy pills.

Step 3: that is If the deviation is .1 gram more, it is the 1st bottle that has heavy pills, it is because only 1 pill is picked from bottle 1 which will add 0.1-gram extra weight with 55.1 gms. 

similarly, If the deviation is .2 grams more, then 2nd bottle has heavy pills, it is because 2 pills picked from bottle 2 will add 0.2-gram extra weight with 55.3 gms. 

if it is .3 gram more then 3rd bottle has heavy pills, if it is .4 gram more then 4th bottle has heavy pills, and so on and 

finally, if the total weight is 56 gms then the 10th bottle has heavy pills.  it is because 10 pills picked from bottle 10 will add 1 gram extra weight with 56 gms. 

In this way we can find the bottle with heavy pills logically, using these simple steps.