**Simple Problem**

There are 12 coins, and one coin has less weight. We have to find the odd coin by measuring only three times.

**Complex Problem**

There are 12 coins, and one coin is of different weight (more or less). We have to find the odd coin by measuring only three times.

**Solutions**

**Simple Problem**

Put 6 coins on one side and other 6 coins on another side.

Take the 6 coins of less weight, and put 3 coins on one side and another 3 coins on another side.

Take the 3 coins of less weight. Put one coin on one side and another coin on another side.

Whichever is of less weight, that is the odd coin. If both are equal, the third coin is of odd coin.

**Complex Problem**

**Incomplete Solution**

Split the 12 coins into three parts, A, B and C, and name them like A1, A2, A3, A4, B1, B2, B3, B4, C1, C2, C3 and C4.

- Put 4 coins of A and 4 coins of B.

- If they are equal, put 2 coins of A on one side, and put C1 and C2 on the other side, and measure the weight.

- If they are not equal, Put A1 and C1 and measure the weight.

- If they are not equal, C1 is the odd coin. Otherwise, C2 is the odd coin.

- If they are equal, put 1 coin of A and C3 and measure the weight.

- If they are equal, C4 is the odd coin. Otherwise, C3 is the odd coin.

- If they are not equal, Put A1 and C1 and measure the weight.

- If they are not equal, remember which one is of less weight. Put C1, C2, C3, C4 on one side and A1, A2, B1, B2 on another side.

- If they are not equal, check whether AB is of more weight or less weight. If it is more weight, then in the previous step, whichever one has more weight that has the odd coin. Same thing applies, if it is of less weight. Let's say, set A has the odd coin. Measure C1 and A1.

- If they are equal, A2 is the odd coin. Otherwise, A1 is the odd coin.

- If they are equal,
**Cannot find the solution**

- If they are not equal, check whether AB is of more weight or less weight. If it is more weight, then in the previous step, whichever one has more weight that has the odd coin. Same thing applies, if it is of less weight. Let's say, set A has the odd coin. Measure C1 and A1.

- If they are equal, put 2 coins of A on one side, and put C1 and C2 on the other side, and measure the weight.

**Complete Solution**

Split the 12 coins into three parts, A, B and C, and name them like A1, A2, A3, A4, B1, B2, B3, B4, C1, C2, C3 and C4.

- Put A1, A2, A3 and A4 on one side and B1, B2, B3 and B4 on the other side.

- If both are equal, put A1, A2, A3 on one side and C1, C2 and C3 on another side.

- If both are equal, C4 is the odd coin. By weighing with any other coin, we can find whether it is heavier or lighter.

- If A is heavier, it means one of C1, C2 and C3 is the lighter odd coin. Weigh C1 and C2. Whichever is of less weight, that is the odd coin. If both are equal, C3 is the lighter odd coin.

- If C is heavier, it means one of C1, C2 and C3 is the heavier odd coin. Weigh C1 and C2. Whichever is of more weight, that is the odd coin. If both are equal, C3 is the heavier odd coin.

- If both are equal, C4 is the odd coin. By weighing with any other coin, we can find whether it is heavier or lighter.
- If A is heavier, put B1, B2, B3 and A4 on one side and C1, C2, C3 and B4 on another side.

- If both are equal, One of A1, A2, A3 is heavier. Weigh A1 and A2. Whichever is heavier, that is the odd coin. If both are equal, A3 is the odd coin.

- If B1, B2, B3 and A4 is still heavier. It means, either A4 is heavier or B4 is lighter. Weigh A4 with any other coin (other than B4). If it is equal, then B4 is the odd coin of less weight. Otherwise, A4 is the odd coin of more weight.

- If B1, B2, B3 and A4 is lighter, then one of B1, B2 and B3 is the lighter odd coin. Weigh B1 and B2. Whichever is lighter, that is the odd coin. If both are equal, B3 is the lighter odd coin.

- If both are equal, One of A1, A2, A3 is heavier. Weigh A1 and A2. Whichever is heavier, that is the odd coin. If both are equal, A3 is the odd coin.
- If B is heavier, do same as above. Put C1, C2, C3 and A4 on one side and A1, A2, A3 and B4 on another side and proceed the same way.

- If both are equal, put A1, A2, A3 on one side and C1, C2 and C3 on another side.

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