Q. 5 pirates of different ages have a treasure of 100 gold coins.
On their ship, they decide to split the coins using this scheme:
The oldest pirate proposes how to share the coins, and all pirates remaining will vote for or against it.
If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the pirates that remain.
Assuming that all 5 pirates are intelligent, rational, greedy, and do not wish to die, (and are rather good at math for pirates) what will happen?

The Solution . . .

Q. 1. A rich merchant had collected many gold coins. He did not want anybody to know about them. One day, his wife asked, "How many gold coins do we have?"
After pausing a moment, he replied, "Well! If I divide the coins into two unequal numbers, then 39 times the difference between the two numbers equals the difference between the squares of the two numbers."
The wife looked puzzled. Can you help the merchant's wife by finding out how many gold coins they have? 


Answer
39

Solution:

The merchant has 39 gold coins.
It is easy to check this... Let's divide the 39 coins into two unequal numbers, say, 33 and 6. Then,

39 (33 − 6) = (33 × 33) − (6 × 6).

Q. The distance between Station Atena and Station Barcena is 78 miles. A train starts from Atena towards Barcena. A bird starts at the same time from Barcena straight towards the moving train. On reaching the train, it instantaneously turns back and returns to Barcena. The bird makes these journeys from Barcena to the train and back to Barcena continuously till the train reaches Barcena. The bird finally returns to Barcena and rests. Calculate the total distance in miles the bird travels in the following two cases: 1. the bird flies at 70 miles per hour and the speed of the train is 60 miles per hour 
2. the bird flies at 60 miles per hour and the speed of the train is 70 miles per hour  


Ans. 1.  
91
Ans 2
72

Solution:
Case 1: Bird flies at a speed greater than that of the train
The train (at a speed of 60 miles per hour) travels 60 miles in 60 minutes.
Therefore, the train travels from Atena to Barcena (78 miles) in 78 minutes.
Importantly, the bird makes the journeys continuously back and forth for this same amount of time (namely, 78 minutes).
Thus, the total distance traveled by the bird
= 70 miles per hour × 78 minutes = 70 × 78 / 60 miles = 91 miles.

Case 2: Bird flies at a speed less than that of the train
In 36 minutes, the bird travels 36 miles, the train travels 42 miles, and the two meet.
Now, the train (which is traveling at a speed greater than that of the bird) will reach Barcena before the bird.
So, the bird simply returns to Barcena (a return journey of 36 miles).
Thus, the total distance traveled by the bird is 72 miles.
 

Q. Last vacation, my cousin came over to stay at my home. We made the most of her stay at my place... and I even earned a few chocolates.
Everyday, we would play a game of chess. Whoever lost the game owed a chocolate to the other. After the last game we played (that was the day she was to leave), we counted the number of games each of us had won and lost. Wow! I had won more than her. So, she handed me 21 chocolates... though she herself was the winner in 6 games.
How many days did my cousin spend at my place? 


Answer
33
Solution:
My cousin won 6 games. Since I got 21 chocolates, I must have won 21 games more than my cousin did. So, I won a total of 27 games.
Thus, the total number of games that we played was 33. Since we played a game each day, that was the number of days my cousin stayed at my house!

Q. It was Maria's first day at school. The teacher suggested that it would be a good idea for each child to meet every other child in the class. The teacher said, "When you meet, please shake hands and introduce yourself by name."
If there were 11 children in the class, how many total handshakes were there?
 


Answer
55
Solution:
The class has 11 children. The first child shakes hands with the other 10 children. The second child has already shaken hands with the first child, and so has to shake hands with only the other 9 children. In this manner, the second-last child has to shake hands with only one child, and the last child has already met all the children. Thus, the number of handshakes is
10 + 9 + ........ + 2 + 1 = 55.
If there were 11 children in the class, then there were 55 total handshakes.

Q. Can you find a quick and elegant way to add the numbers from 1 to 28 ?
In other words, what is the sum you obtain in the following case:
1 + 2 + 3 + ....... + 28 =?


Answer
406

Solution:
Here's an elegant way:
1 + 28 = 29
2 + 27 = 29
3 + 26 = 29
.....................
.....................
14 + 15 = 29
Adding the above equations gives
1 + 2 + 3 + ....... + 28 = 14 × 29 = 406

Q. Haretown and Tortoiseville are 40 miles apart. A hare travels at 7 miles per hour from Haretown to Tortoiseville, while a tortoise travels at 3 miles per hour from Tortoiseville to Haretown.
If both set out at the same time, how many miles will the hare have to travel before meeting the tortoise en route? 


Answer
28

Solution:
The hare and the tortoise are together covering the distance at 10 miles per hour (i.e., on adding their speeds).
So, they will cover the distance of 40 miles in 4 hours.
Thus, in 4 hours, they will meet and the hare will have traveled 28 miles.  
Alternative Solution through Equations:
Note that : Distance = Speed × Time
Let t be the time before the hare and the tortoise meet.
In t hours, the hare will travel 7 t miles.
In t hours, the tortoise will travel 3 t miles.
Now,
7 t + 3 t = 40
So, t = 40 ⁄ 10 = 4 hours.
Thus, distance traveled by hare before meeting = 7 × 4 = 28 miles