Assume that you have 8 identical-looking coins and a two-pan balance scale with
ID: 3848324 • Letter: A
Question
Assume that you have 8 identical-looking coins and a two-pan balance scale with no weights. One of the coins is fake, but it is not known whether it is lighter or heavier than the real 7 coins. Describe your idea to determine in the minimum number of weighings whether the fake coin is lighter or heavier than the others. Present the minimum number of weighings and your answer clearly. In the problem, remember that I do not ask you to find the fake coin among 8 coins using the scale. You don’t need to find out which coin is fake. The question is that you should be able to identify whether the fake coin is heavier or lighter than the real coins.
Explanation / Answer
Make 2 Heaps of 4 coins each. Heap A and Heap B. Now in the scale, weigh them..
Lets assume, Heap A weighs more and Heap B weighs less.. (This can se dafely assumed, as due to the fake coin, they can't be of same weight, hence they must be less or heavy)
Now Remember Heap A is heavy.
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Now the problem is to identify, which out of A and B is erroneous, if A is erroneous, then fake coin is heavy, otherwise if heap B is erroneous, then fake coin is lighter.
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How to find erroneous heap:
Now, take Heap B, make 2 heaps again out of Heap B, lets say B1 and B2.. Now weigh B1 and B2.. if the weight of B1 and B2 is same.. means all 4 coins of Heap B are of same wieight(Because if there would have been one fake coin in Heap B,then B1 and B2 can't weigh same), and thus Heap B is not errorneous. => Heap A contains fake coin
Instead, if heap B1 and B2 didn't weigh equal, it means the fake coin is among them Hence Heap B contains the fake coin.
Once we determine the heap A or B with fake coin, we can easily say that wheter coin was heavy or lighter.