There are nine identical balls, one of which is heavier than the other eight. What is the least number of weighings, using a two-pan balance, needed for definitely identifying the heavier ball?
Heavier Ball Weighing Puzzle Solution
The problem asks for the minimum number of weighings required to identify a single heavier ball among nine otherwise identical balls, using a two-pan balance.
A two-pan balance allows us to compare the weight of items placed on its pans. There are three possible outcomes when weighing: the left pan is heavier, the right pan is heavier, or both pans are of equal weight.
Weighing Strategy
To find the single heavier ball efficiently, we need a strategy that eliminates as many possibilities as possible with each weighing. The optimal approach involves dividing the balls into groups.
First Weighing
Divide the nine balls into three equal groups:
- Group 1: Balls 1, 2, 3
- Group 2: Balls 4, 5, 6
- Group 3: Balls 7, 8, 9
Place Group 1 (3 balls) on the left pan of the balance and Group 2 (3 balls) on the right pan.
There are three possible outcomes from this first weighing:
- Outcome A: The balance stays level. This means Group 1 and Group 2 have equal weight. The heavier ball must be in the group not weighed, which is Group 3.
- Outcome B: The left pan goes down. This indicates that Group 1 is heavier than Group 2. The heavier ball must be within Group 1.
- Outcome C: The right pan goes down. This indicates that Group 2 is heavier than Group 1. The heavier ball must be within Group 2.
After this first weighing, you have successfully narrowed down the location of the heavier ball to a specific group of 3 balls.
Second Weighing
Take the group of 3 balls that was identified in the first weighing as containing the heavier ball. Let's call these three balls A, B, and C.
Place Ball A on the left pan and Ball B on the right pan.
There are three possible outcomes from this second weighing:
- Outcome X: The balance stays level. This means Ball A and Ball B are of equal weight. The heavier ball must be the one not weighed, which is Ball C.
- Outcome Y: The left pan goes down. This indicates that Ball A is heavier than Ball B. The heavier ball is Ball A.
- Outcome Z: The right pan goes down. This indicates that Ball B is heavier than Ball A. The heavier ball is Ball B.
After this second weighing, you have definitively identified the single heavier ball among the original nine.
Minimum Weighings Required
Using this strategy, we were able to isolate the heavier ball in all possible scenarios with just two weighings. Therefore, the least number of weighings needed to definitely identify the heavier ball among nine is two.
Important Questions from Miscellaneous
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A railway wagon (open at the top) of mass M1 is moving with speed v1 along a straight track. As a result of rain, after some time it gets partially filled with water so that the mass of the wagon becomes M2 and speed becomes v2. Taking the rain to be falling vertically and the water stationery inside the wagon, the relation between the two speeds v1 and v2 is :
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1. Distance between the longitudes becomes zero on North Pole and South Pole.
2. Distance between the longitudes is maximum on the Equator.
3. Number of longitudes is more than number of latitudes.
Which of the statements given above is/are correct?
One block of 2⋅0 kg mass is placed on top of another block of 3⋅0 kg mass. The coefficient of static friction between the two blocks is 0⋅2. The bottom block is pulled with a horizontal force F such that both the blocks move together without slipping. Taking acceleration due to gravity as 10 m/s2, the maximum value of the frictional force is :
