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Process To Identify Fake Coin

**What was the task given? **
If we knew, the fake coin is **lighter or heavier** than original one then the process would have been pretty simple like **this!** But we don't know.
Let's **number** the coins from 1 to 12. We'll make 3 groups of these coins as **1,2,3,4** in one group, **5,6,7,8** in other group and **9,10,11,12 **in one more group.
First of all weigh **1,2,3,4** against** 5,6,7,8.**
**CASE 1 : 1,2,3,4 = 5,6,7,8**
That means coin among **9,10,11,12** is fake one. So weigh **9,10** against **11,8.**
**C****ASE 1.1 : **If **9,10 = 11,8** then **12** is fake coin.
**CASE 1.2 :** If **9****,10 > 11,8** then either **9** or **10** is** heavier** (hence fake) or **11** is** ****lig****hter** (hence fake). Weigh **9** against **10.** If they balance then **11** is fake one. If they don't then heavier of **9** & **10** is fake**.**
**C****ASE 1.3 :** If **9****,10 < 11,8** then either **9** or **10** is** lighter** (hence fake) or **11** is **heavier** (hence fake). Weigh **9** against **10.** If they balance then **11** is** ****fake** one. If they don't then lighter of **9** & **10** is fake.
**CASE 2 : 1,2,3,4 < 5,6,7,8 **
This means coins **9,****10,11,12** are **real** ones. So weigh **1,2,5** against **3,6,9****.** Why these particulars you will know in the process.
** CASE 2.1 : 1,2,5 = 3,6,9**
Indicates that either** 7** or **8** is **heavy** of** 4** is** lighter.** So weight 7 against 8. If they **balance**, then** 4** is **fake** one. If they don't then **heavier** of **7** & **8** is fake.
**CASE 2.2 : 1,2,5 < 3,6,9**
Now** 5** can't make this one** light** of **3** can't make it **heavy** since 1,2,3,4 < 5,6,7,8. Hence, either **1** or **2** is** li****ghter** or **6** is **hea****vier** (9 is perfect one). Next weigh 1 against 2. If they **balance** that means **6** is** ****hea****vier** & hence** **fake one. If they don't balance then that means 6 is original one & lighter of **1 **& **2** is** **fake.
** CASE 2.3 : 1,2,5 > 3,6,9**
Here **1,2** can't make this **heavier** or **6** can't make it **lighter** as 1,2,3,4 < 5,6,7,8. Hence either **3** must be **lighter** or **5** could be **heavier.** There is no way that 3 & 5 will balance. So skipping this, directly testing **3** or** 5** against any **good coin** say 11.
** CASE 2.3.1** : If **3**** = ****11** then **5** is fake one.
**CASE 2.3.2** : If** 3 < 11** then **3** is fake one. **3 > 11** is impossible as we already deduced that 3 is either lighter one or real one.
**CASE 3 :** On the similar note, we can deduce fake coin if **1,2,3,4 ****> 5,6,7,8.** The same is depicted in the chart below.
To conclude, we need to use balance **only 3 times** (count number of times it is used in each case) to know the fake coin.

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