**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**

Now

**5**can't make this one

**light**of

**3**can't make it

**heavy**since 1,2,3,4 ,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 ence 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 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.