How will you find out that which coin is heavier or which one is lighter ?

Question

We have 12 coins and a balance. 11 coins are of the same weight, but one coin differs in weight (note that you do not know whether the coin with different weight is heavier or lighter!). You may perform three weighings to find out which coin has a different weight, and whether this coin is heavier or lighter.

How should you perform these three weighings to find out which coin has a different weight, and whether this coin is heavier or lighter?

Answer 1

We will try to arrange each use of the balance in such a way that the answers are divided into three equal (or nearly equal) groups. So the first weighing should be four coins in the left cup and four coins in the right cup. If the left cup weighs less than the right cup, then we have 8 potential outcomes for the right answer: the fake coin is one out of the four on the left, and it is lighter, or the fake coin is one out of the four on the right, and it is heavier. Similarly, we have 8 potential outcomes in each of the other two cases.

Suppose the left cup is lighter than the right cup. Let us denote the left coins as coins 1, 2, 3, and 4 and the right coins as 5, 6, 7, and 8. For the next step, we should divide the 8 potential answers into groups of 3, 3 and 2. For example, we want to have three answers in the case when the left cup is lighter or equal to the right cup, and 2 answers when the left cup is heavier than the right cup. That means that 3 out of the 8 coins should be left in the same place that they started, 3 should be taken away, and 2 should change places. We may use coins 9, 10, 11, and 12 to supplement each weighing, in order to have an equal number of coins in each cup. For example, we can leave coins 1, 2, and 5 in their places, coins 3, 4 and 6 can be removed from the balance, and coins 7 and 8 can change cups. In this case, we need to add three good coins, so our second weighing is: 1, 2, 7, and 8 in the left cup; and 5, 9, 10, 11 in the right cup. If the cups are equal, then the fake coin will be found among 3, 4 or 6. If the left cup is lighter, then the fake coin is among 1, 2, and 5, and if the left cup is heavier, then the fake coin is among 7 or 8, and for each number we know if it is heavier or lighter.

Answer 2

First - Divide into two sections - 6 each and you can find which of the sixes has heavier coin

Second - Divide the 6 coins heavier into 3 each and find the heavier 3

Third- Take two coins out of the heavier three and weigh them against each other. If two are equal , then its the one that you haven't weighed , else the heavier one on the balance

Comments

We don't know if differed coin is heavior or lighter...so this solution will not work. Would you like to edit the solution

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Would have liked to edit the answer :) Guess its a little late , Thanks a ton sir.

Answer 3

There are several ways to solve this problem. An elegant solution is shown below. Note again that in advance, you do not know whether the coin with different weight is heavier or lighter!

Number the coins from 1 up to 12. Perform the following three weighings:

Left side:  Right side:
Weighing 1: 1  2  3  10 4  5  6  11
Weighing 2: 1  2  3  11 7  8  9  10
Weighing 3: 1  4  7  10 3  6  9  12

Call the outcome of a weighing "L" if the left side is most heavy, call the outcome "R" if the right side is most heavy, and call the outcome "B" if the left and right sides have the same weight.

Then the following outcomes are possible:

Weighing 1: Weighing 2: Weighing 3: Different coin:
L   L   L   1 heavier
L   L   R   3 heavier
L   L   B   2 heavier
L   R   L   10 heavier
L   R   B   11 lighter
L   B   L   6 lighter
L   B   R   4 lighter
L   B   B   5 lighter
R   L   R   10 lighter
R   L   B   11 heavier
R   R   L   3 lighter
R   R   R   1 lighter
R   R   B   2 lighter
R   B   L   4 heavier
R   B   R   6 heavier
R   B   B   5 heavier
B   L   L   9 lighter
B   L   R   7 lighter
B   L   B   8 lighter
B   R   L   7 heavier
B   R   R   9 heavier
B   R   B   8 heavier
B   B   L   12 lighter
B   B   R   12 heavier

Note: outcomes LRR, RLL, and BBB are not possible.