Question

PUZZLES
1) You have two strings whose only known property is that when you light one end of either string it takes exactly one hour to burn. The rate at which the strings will burn is completely random an each string is different. How do you measure 45 minutes?

2) Two friends decide to get together; so they start riding bikes towards each other. They plan to meet halfway. Each is riding at 6 MPH. They live 36 miles apart. On of them has a pet carrier pigeon and it starts flying the instant the friends start travelling. The pigeon flies back and forth at 18 MPH between the 2 friends until the friends meet. How many miles does the pigeon travel.

3) A king wants his daughter to marry the smartest of 3 extremely intelligent young princes, and so the king's wise men devised an intelligence test. The princes are gathered into a room and seated, facing one another, and are shown 2 black hats and 3 white hats. They are blindfolded, and 1 hat is placed on each of their heads, with the remaining hats hidden in a different room. The king tells them that the first prince to deduce the colour of his hat without removing it or looking at it will marry his daughter. A wrong guess will mean death. The blindfolds are then removed. You are one of the princes. You see 2 white hats on the other prince's heads. After some time you realize that the other prince's are unable to deduce the colour of their hat, or are unwilling to guess. What colour is your hat?

Answer 1

1)

Fold the 1st string to have 4 folds (fold it twice in the middle). Its length becomes 1/4 th. Light both strings.  Start measuring after the folded string burns completely, it will be 45 minutes when the second strings completely burns.

2)

Let us say the first friend F1 starts at A and the other friend F2 starts off at B.  
AB = 36 miles.    F1 travels to right toward B at 6 miles/hr  and F2 travels towards A at 6 miles/hr.  Pigeon starts from A towards B at 18 mph.

Relative velocity of F1 and F2 is = 6+6 = 12 mph. So they cover the distance AB in 36miles/12 mph = 3 hours. 

So the pigeon flies in 3 hours = 3 * 18 mph = 54 miles.

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Alternately, the long method:

Relative velocity of Pigeon and F2 = 18 + 6 = 24 mph.  They have to a total of 36 miles to meet.  So, they meet after 36 miles / 24 mph = 1.5 hours = 90 min.

In 90 minutes F1 has moved to a point C, 6*1.5 = 9 miles from A  
    and F2 has moved to a point D at a distance 9 miles from B or 27 miles from A.

Now the pigeon starts flying towards F1 and distance between F1 & pigeon is 18 miles.

So they take 18 miles/24 mph = 0.75 hr to meet.
By now A moves to a point E at 4.5 miles from C or 13.5 miles from A.  Now F2 is at F, 13.5 miles from B or 22.5 miles from A. The distance between F1 and F2 is 9 miles.

Now pigeon flies towards F2 and meets F2 in 9/24 = 3/8 hrs.  By now, F1 travels 3*6/8 = 2.25 miles to a point G at 15.75 miles from A and F2 is at 20.25 miles from A.  The distance between them is 4.5 miles.

Now pigeon and F1 meet after 4.5/24 hrs. In this time F1 and F2 cover a distance of 2.25 miles.  So distance between F1 and F2 becomes 2.25 miles. 

The distance traveled by pigeon = 18 mph * [ 36 + 18 + 9 + 4.5 + 2.25 + .... ] / 24
                          = 18 * 36 * [ 1 + 1/2 + 1/4 + 1/8 + 1/16 + .... ] / 24
                          =  18 * 36 * 2 / 24 = 54 miles

C) 

Answer 2

1. Let the two ropes be A and B. Burn A from both sides and B from one side simultaneously. A will be burnt out in half an hour. At that point B has been burnt half. Now burn the other side of B. Now it will finish in another 15 minutes. So when B is burnt off, 45 minutes are over.

2. The two friends start their journey at 6mph towards each other. Since their speed are equal, they will travel same distances and it is (36/2 =18 mile each). Time taken for two friends to meet = 18/6 =3hours. In three hours the pigeon will fly (18X3 =54miles).

3.Let three princes be A, B and me. A has white hat, B has white hat and I don't know what I have.
Let me assume that I have a black hat. Then A would see a black and a white hat. Then he would guess that he has white hat suddenly. (Because A knows the king would not use 2 black hats. If he uses 2 black hats, the third person will easily tell that he has white hat.) 
But that is not the case. A is not able to guess and confused. So is B. 
So the other option is that I have a white hat. It will confuse every one and no one can deduce directly. 
I deduced because I am more intelligent and deserve the Princess. :P