Puzzle : The Case of Missing Servant

A king has 100 identical servants, each with a different rank between 1 and 100. At the end of each day, each servant comes into the king's quarters, one-by-one, in a random order, and announces his rank to let the king know that he is done working for the day. 

For example, servant 14 comes in and says "Servant 14, reporting in." 

One day, the king's aide comes in and tells the king that one of the servants is missing, though he isn't sure which one.

Before the other servants begin reporting in for the night, the king asks for a piece of paper to write on to help him figure out which servant is missing. 

Unfortunately, all that's available is a very small piece that can only hold one number at a time. The king is free to erase what he writes and write something new as many times as he likes, but he can only have one number written down at a time. 

The king's memory is bad and he won't be able to remember all the exact numbers as the servants report in, so he must use the paper to help him.

How can he use the paper such that once the final servant has reported in, he'll know exactly which servant is missing?

Mathematical Trick to know the missing servant! 

Solution : The Missing Servant in the Case

What was the case?

When the first servant comes in, the king should write his number on the small piece of paper. For every next servant that reports in, the king should add that servant's number to the current number written on the paper, and then write this new number on the paper while erasing old one.

Addition of numbers from 1 to 100 = 5050.

Hence, 

Missing Servant Number = 5050 -  Addition of ranks of 99 Servants.

So, depending on how far the addition of 99 servants' rank goes to near 5050, the king can easily deduce the rank of missing servant.

For example, if the addition that king has after 99 servants report in is 5000 then the servant having rank = 5050 - 5000 = 50 must be missing. 

 

Arrange Positions Around The Round-Table

King Arthur and his eleven honorable knights must sit on a round-table. In how many ways can you arrange the group, if no honorable knight can sit between two older honorable knights?

Here are the possible combinations!

Source 

Possible Positions Around The Round Table

What was the challenge?

If king K is sitting at the center then the youngest knight must sit to right or left of the king i.e. 2 possible positions for him.

The second youngest knight now can sit either left or right of the group of 2 made above.

The third youngest knight now can sit either left or right of the group of 3 made above.

And so on.

That is every knight has 2 possible positions except the oldest knight who will have only 1 position left.

This will be make sure each of the knight (except youngest one) has at least 1 younger neighbor (youngest one has king as one neighbor).

So after putting youngest in 2 possible ways the next youngest can be put another 2 possible ways. That is 4 possible combinations for 2.

Similarly, for arranging 3 knights' positions there can be 2^3 = 8 possible combinations.

This way for 10 knights (excluding the oldest which will have only 1 seat available at the end) there are 2^10 = 1024 possible combinations.

"Who is telling the truth?"

King Octopus has servants with six, seven, or eight legs. The servants with seven legs always lie, but the servants with either six or eight legs always say the truth.

One day, 4 servants met :

The blue one says: “Altogether we have 28 legs”;

the green one says: “Altogether we have 27 legs”;

the yellow one says: “Altogether we have 26 legs”;

the red one says: “Altogether we have 25 legs”.

 
What is the color of the servant that says the truth?

"I'm telling the truth and my color is..." 

"Listen to me; I'm telling the truth!"

What others making statements?

Since, each of 4 servants telling different numbers then only 1 of them must be telling the truth & other must be lying.

In that case, all 3 must be having 7 legs i.e. 21 legs while the one who is making true statement must have either 6 or 8 legs. Therefore, there must be 21 + 6 = 27 or 21 + 8 = 29 legs altogether.

Nobody is saying that they altogether have 29 legs but the green octopus is saying that they altogether have 27 legs. Hence, the green octopus must be telling the truth. 

CheckMate The Opponent in 2 Moves!

Can White checkmate Black in two moves from this position?

Here are those 2 moves! 

2 Moves to CheckMate the Opponent

Where the game stands?


1.Move Rook to b3 i.e. perform Rb3.

2.That will force opponent to move king to a5 i.e. execute Ka5.

3. Finally,now move Rook to a3 (Ra3) and Checkmate the opponent.

Truth or Lie Puzzle

Out of three people (Lavesh, Mayank and Manoj), one of them is a king, one a bureaucrat, and one a thief.
The king always tells the truth, the bureaucrat always lies, and the thief can either lie or tell the truth.
Lavesh says: 'Manoj is a bureaucrat.'
 
Mayank says: 'Lavesh is a king.'
 
Manoj says: 'I am the thief.'
Who is the king, who the bureaucrat, and who the thief? 

Find the TRUTH here! 
 

King, Bureaucrat and Thief

What was the story? 

Let's take a look at the statements made by 3.

Lavesh says: 'Manoj is a bureaucrat.'
Mayank says: 'Lavesh is a king.'
Manoj says: 'I am the thief.'

Now let's test truth in each statement.

If we assume Mayank is king and telling the truth than according to him Lavesh is also king. But that's not the possible case.

Now if we assume Manoj is king then his statement must be true where he is saying that he is thief. So how he can be a king & thief at the same time.

Hence, the Lavesh must be a king. Mayank is here telling the truth and hence must not be bureaucrat and hence he is thief. So Manoj is bureaucrat lying in his statement.
 

 

Wise Men In Survival Game

A stark raving mad king tells his 100 wisest men he is about to line them up and that he will place either a red or blue hat on each of their heads.

Once lined up, they must not communicate among themselves. Nor may they attempt to look behind them or remove their own hat.The king tells the wise men that they will be able to see all the hats in front of them. They will not be able to see the color of their own hat or the hats behind them, although they will be able to hear the answers from all those behind them.

The king will then start with the wise man in the back and ask "what color is your hat?" The wise man will only be allowed to answer "red" or "blue," nothing more. If the answer is incorrect then the wise man will be silently killed. If the answer is correct then the wise man may live but must remain absolutely silent.The king will then move on to the next wise man and repeat the question.
 
The king makes it clear that if anyone breaks the rules then all the wise men will die, then allows the wise men to consult before lining them up. The king listens in while the wise men consult each other to make sure they don't devise a plan to cheat. To communicate anything more than their guess of red or blue by coughing or shuffling would be breaking the rules.

What is the maximum number of men they can be guaranteed to save?

Almost all can survive! Click here to know! 

Source 

Master Plan By Wise Men

Why this master plan needed? 

99 can be guaranteed to save! How?

Even if the person behind calls out the color of the hat that next person is wearing both would be survived only if they are wearing same color of hat. 

So how 99 can be saved?

For a simplicity, let's assume there are only 10 wise men & (only) assume we are among them. Now, we need to make a master plan to survive from this game of death.

One of us need to agree to sacrifice his life to save 9 of us & this person would be the first one in line. He will be survived of he has good luck.

The first person in line should shout RED if he founds number of RED hats even otherwise he should shout BLUE. Now if he has good luck then the hat color of his own hat would match & he would be survived.

The clue given by the first person is very important. Right from second person everyone need to count number of RED hats in front of him. Additionally, the next person need to keep track of number of RED hats that people behind him are wearing.

Identify The Cards

From a pack of 52 cards ,I placed 4 cards on the table.

I will give you 4 clues about the cards:

Clue 1: Card on left cannot be greater than card on the right.
Clue 2: Difference between 1st card and 3rd card is 8.
Clue 3: There is no card of ace.
Clue 4: There is no face cards (queen,king,jacks).
Clue 5: Difference between 2nd card and 4th card is 7.

Identify four cards ?

Cards identified here!

Source 

Identified Cards From Clues

What was the task given? 

Let's list the clues once again here for our convenience.

Clue 1: Card on left cannot be greater than card on the right.
Clue 2: Difference between 1st card and 3rd card is 8.
Clue 3: There is no card of ace.
Clue 4: There is no face cards (queen,king,jacks).
Clue 5: Difference between 2nd card and 4th card is 7.

From Clue 4, it's very clear that there is no King, Queen or Jack card.

From Clue 2 & Clue 3, we have combinations of either 1,9 or 2,10 at first & third place. But Clue 3 eliminates the combination of 1,9.So at first place we have 2 & at third we have 10.

Again from Clue 5 & Clue 3, possible combinations at second & fourth place are 2,9 & 3,10. If it was 2,9 then 4 cards would have been like 2,2,10,9. But according to Clue 1 the card on left can't be greater than that at right. Here, the card at third place (10) is greater than that at fourth place (9) placed at right. Hence, this would be invalid combination.

Hence the correct combination for the second & fourth place is 3,10.

So we have 4 cards as 2,3,10,10.