"Are you holding true or fake coin?"

You have 101 coins, and you know that 50 of them are counterfeit. Every true coin has the same weight, an unknown integer, and every false coin has the same weight,which differs from that of a true coin by 1 gram. You also have a two-pan pointer scale that will show you the difference in weight between the contents of each pan. You choose one coin. 

Can you tell in a single weighing whether it’s true or false?

Well, this trick will help you to identify that coin! 

Knowing The Truth of the Coin in Hand!

What was the task given?

Yes, you can tell that whether the coin is true or false with single weighting.Just divide 100 coins into 2 groups of 50 coins each & put into 2 pans of weighing balance.

Let's assume true coin weighs 1 gram (or 2 gram) & fake coin weighs 2 gram (or 1 gram). Remember, if the sum of 2 integers is even then difference between two is bound to be even. And if the sum of those is odd then difference between them has to be odd.

CASE 1 :

If the coin that you are holding is true then the total weight on the balance will be
50 + (50x2) = 50 + 100 = 150  (or 50x2 + 50 = 150). So, the total sum of weights in 2 pans is even, hence difference between them has to be even. For example, if those 150 grams are distributed as 80 vs 70 then difference between them is 10 which is even.


CASE 2 :

If the coin you are holding is fake then the total weight on the balance will be
51 + (49x2) = 51 + 98 = 149 (or 51x2 + 49x1 = 153).

Here, total is odd hence the difference must be odd too. For example, if above 149 grams are distributed as 90 vs 59 then pointer of balance will point at 31 which is odd.

Conclusion:

In short, you have to notice the difference between 2 weights on the pans. If it's even then the coin you are holding is true and if difference is odd then you are holding a fake coin.
 

The Poisonous Glass?

You are given 4 identical glasses, completely filled with transparent, odorless liquids. Three of the liquids are pure water, and the fourth is poison, which is slightly heavier. If the water glasses weigh 250 grams each, and the poisoned glass weighs 260 grams, how can you figure out which one is which, using a measuring scale just once?

Here is the process to identify the poisonous glass!

Identifying The Poisonous Glass!

What was the challenge?

Let's number the 4 glasses as 1,2,3 and 4.

Here are steps to identify the glass with poisonous liquid in it. 

1. Empty the glass 1 into another empty glass.

2. Take about 1/4th liquid from glass 2 and pour it into emptied glass 1.

3. Take 3/4th of liquid from glass 3 and pout it into glass 1.

Now, glass 1 has 1/4th of liquid from glass 2 and 3/4th from glass 3.

4. Now put glass 1 and glass 4 on the measuring scale.

    4.1 - If it weighs exactly 500 gm then it suggests that glass 1 had poison.
  
    4.2 - If it weighs between 500 - 505 grams (precisely 502.5 if exactly 1/4th of liquid was taken from glass 2 to glass 1) then obviously glass 2 is with poisonous liquid.

    4.3 - If it weighs between 505 - 510 grams (precisely 507.5 if exactly 3/4th of liquid was taken from glass 3 to glass 1) then obviously glass 3 is with poisonous liquid.

    4.4 - If it weighs exactly 510 gm then it suggests that glass 4 has poison.

 

Distinguish The Fake Coin

You have twelve coins. You know that one is fake. The only thing that distinguishes the fake coin from the real coins is that its weight is imperceptibly different. You have a perfectly balanced scale. The scale only tells you which side weighs more than the other side.

What is the smallest number of times you must use the scale in order to always find the fake coin?
 
Use only the twelve coins themselves and no others, no other weights, no cutting coins, no pencil marks on the scale. etc.

These are modern coins, so the fake coin is not necessarily lighter.

Presume the worst case scenario and don't hope that you will pick the right coin on the first attempt.

Process to identify the fake one! 

Source 

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.

   CASE 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 lighter (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. 

   CASE 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.