Spot The Liar Among

There is a party of 100 politicians. All of them are either honest or liars. You walk in knowing two things:

1. At least one of them is honest.

2. If you take any two politicians, at least one of them is a liar.

From this information, can you know how many are liars and how many are honest?

Know the total number of liars here! 

Source

Knowing The Number of Liars

What is the question?

The information that we have - 

1. At least one of them is honest.

2. If you take any two politicians, at least one of them is a liar. 

Only one of them must be honest and other 99 must be liars. 

Those are the only numbers that can justify the second piece of information. In the case, if any 2 are selected at least 1 of them liar while other may or may not liar (i.e. may be honest). 

A Brillian Deception

A witch owns a field containing many gold mines. She hires one man at a time to mine this gold for her. She promises 10% of what a man mines in a day, and he gives her the rest. Because she is blind, she has three magic bags who can talk. They report how much gold they held each day, and this is how she finds out if men are cheating her. 

Upon getting the job, each man agrees that if he isn't honest, then he will be turned into stone. So around the witch's mines, many statues lay! 

Now comes an honest man named Garry. He accepts the job gladly. 

The witch, who didn't trust him said, "If I wrongly accuse you of cheating me, then I'll be turned into stone."  

That night, Garry, having honestly done his first day's job, overheard the bags talking to the witch. He then formulated a plan... 

The next night, he submitted his gold, and kept 1.6 pounds of gold. Later, the witch talked with her bags.

The first bag said it held 16 pounds that day. The second one said it held 5 pounds. The third one said it held 2 pounds. 

Beaming, the witch confronted Garry. "You scoundrel, you think you could fool me. Now you shall turn into stone!" the witch cried. One second later, the witch was hard as a rock, and very grey-looking.

How did Garry brilliantly deceive the witch? 

Here is the Garry's Master Plan!
 

Mathematics Behind The Brilliant Deception!

What was the deception?

As per the honest man, he must have mined 16 Pounds of gold since he kept 1.6 Pounds (10% of 16) gold for himself.

And as per magical bags, since Bag no.1 said it had 16 Pounds, Bag no.2 said 5 Pounds and Bag no.3 said 2 Pounds the total gold mined 16 + 5 + 2 = 23 Pounds.

We know, honest man Garry would never do any fraud and neither of magical bag would lie. 

So, it's clear that some of pounds are counted multiple times by magical bags. There are 23 - 16 = 7 Pounds gold extra found by those bags.

Now, Bag No.3 must have 2 Pounds in real.

And to make count of Bag No.2 as 5, Garry must had put 3 Pound + Bag No.3 itself in Bag No.2. Hence, the Bag No.2 informed witch that it had total 5 Pound of gold.

Finally, to force Bag No.1 to tell it's count as 16, Garry must had put 
11 Pounds + Bag No.2 (5 Pounds = 3 + Bag No.1) = 11 + 5 = 16 Pounds.

In short, Garry put 2 pounds of gold in Bag No.3 and put that Bag No.3 in Bag No.2 where he had already added 3 Pounds of gold. After that, he put this Bag No.2 in Bag No.3 where he had already added 11 Pounds of gold (somewhat like below picture).

 

This way, 2 Pounds of gold in Bag No.3 are counted 2 extra times and 3 Pounds gold of Bag No.2 are counted 1 more extra time. That is 2 + 2 + 3 = 7 Pounds of extra gold found by bags.

Black or White Dot on Forehead

There are five men, let's say Tarun, Harish, Lavesh, Manoj and Manish. All of them have a dot mark in their forehead. They can't see the dot on their own forehead, but can see the ones on others. The owner of WHITE dot is an honest person and will never lie, while the owner of BLACK dot always tell the lie.

This are the statement from Tarun, Harish, Lavesh, and Manish:

Tarun: 'I see 3 whites and 1 black'

Harish: 'I see 4 black'

Lavesh: 'I see 3 black and 1 white'

Manish: 'I see 4 white'

What color is the dot on each Tarun, Harish, Lavesh, Manoj, Manish forehead?

Know color of dot on each forehead!

Source 

Men with White Dots on Foreheads

Read this story first!

Let's take a look at who said what. 

Tarun: 'I see 3 whites and 1 black'
Harish: 'I see 4 black'
Lavesh: 'I see 3 black and 1 white'
Manish: 'I see 4 white'
 
Harish and Manish made very contracting statements; so are statement of Tarun and Lavesh. That means, only one of them is telling the truth. Hence, only one of 4 have WHITE dot on his forehead. Till now we don't have idea of what Manoj had on his forehead. Even if he has WHITE dot then there can be maximum 2 WHITE dots among those 5.    

Truth of Tarun and Manish:

Tarun and Manish must be lying as they are saying that they have seen 3 and 4 WHITE dots respectively. According to our first conclusion, there can be maximum 2 WHITE dots possible.

Truth of Harish:

Next if we assume Harish has WHITE dot and telling the truth then all other must be lying including Manoj and Lavesh. As per Lavesh, he had seen 3 BLACK and 1 WHITE dot. Now he must have seen BLACK dots on foreheads of Manoj, Tarun, Manish and WHITE dot on forehead of Harish as assumed. That mean he is telling truth though he has BLACK dot. But the man with BLACK is always lying, hence this case is also INVALID. 

Truth of Lavesh:

So the only person left is Lavesh and must be telling the TRUTH with WHITE dot on his forehead. Hence, as his statement is suggesting, the Manoj must have WHITE dot and other three Tarun, Lavesh, Harish have BLACK dots (we already concluded these 3 are lying).