## BAG OF BALLS

Q. Mukesh purchased eight balls of the same weight, size and color and kept them in a bag. After some time his younger brother Rakesh came home and put his ball in the same bag. This ball was similar to the ones which Mukesh had put in, but was not equal in weight to them. When Rakesh came for the ball after sometime, he insisted on having his own ball.Now Mukesh faced a problem. He had to sort out Rakesh’s ball. Then it occurred to him that he should use a weighing machine. So, he procured one from his friend Ramesh and started sorting Rakesh’s ball.

Can you help Mukesh in finding the ball that belonged to Rakesh in minimum time? Also state whether his ball was lighter or heavier than Mukesh’s balls?

A. To begin with, we shall divide the 9 balls in 3 heaps of 3 balls each. Suppose the heaps are ‘A’, ‘B’ & ‘C’

**First weighing:**

We will place heap ‘A’ on one pan of the balance and heap ‘B’ on the other. There are three possibilities:

(1) ‘A’ = ‘B’ (2) ‘A’ heavier than ‘B’ and (3) ‘B’ heavier than ‘A’.

**Second weighing:**

Stage 1 – second weighing – ‘A’ and ‘C’ to be put on pans. If ‘A’ is heavier than ‘C’, then Rakesh’s ball is lighter than that of Mukesh’s. And if ‘A’ is lighter than ‘C’, Rakesh’s ball is heavier. At this stage it will come out that Rakesh’s ball is in help ‘C’.

Stage 2 – ‘A’ and ‘C’ will be put on pans. If ‘A’ and ‘C’ are equal then Mukesh’s ball is in help ‘B’ and is heavier. If ‘C’ is heavier than ‘A’, Rakesh’s ball is lighter and is in heap ‘A’.

Stage 3 – ‘B’ and ‘C’ will be put on pans. If ‘B’ and ‘C’ are equal then Rakesh’s ball is heavy and is in heap ‘A’. If ‘C’ is heavier than ‘B’, then, Rakesh’s ball is light and is in ‘B’ heap. Thus, we would come to know the exact heap where Rakesh’s ball exists and whether it is heavy or light in weight in weight in comparison with Mukesh’s ball.

**Third weighing:**

In the third weighing experiment, we take any two balls from Rakesh’s heap. Now, if these two weigh equal, the third ball belongs to Rakesh. If the scale is uneven, the raised pan has Rakesh’s ball because it (the ball) is lighter and its pan will lift up. In case the ball is heavy, it is in the lower pan of the balance. Weighing these balls three times we find out Rakesh’s ball and also know whether it is lighter or heavier than Mukesh’s.

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