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Analysing Interesting Circular Race Track


What was the interesting fact about?

Just imagine that each car carries a flag on it and on meeting pass on that flag to the next car. Obviously, this flag will move at the constant speed around the track as cars carrying it are also moving at the constant speed. So, the flag will be back at the original position after 1 hour.

Let's assume there are only 2 cars on the track at diagonally opposite points as shown below. 

Analysing Interesting Circular Race Track


After 15 minutes, on meeting with Car 2, Car 1 will pass on the flag & both will reverse their own direction. 30 minutes later (i.e. 45 minutes after start) both cars again meet each other and Car 2 will pass on flag back to Car 1 & directions are reversed again. Again in another 15 minutes (i.e. after 1 hour from start), both cars are back at the original positions. 

Now, let's suppose that there are 4 cars on the track positioned as below.

Analysing Interesting Circular Race Track


The above image shows how cars will be positioned after different points of time & how they reverse direction after meeting.

Again, all are back to the original position after 1 hour including the flag position. One more thing to note that the orders in which cars are never changes. It remains as 1-2-3-4 clockwise. 

To conclude, for 'n' number of cars, at some point of time all the cars will be in original positions in future.   
 

Geometrical Puzzle

Find the area of the shaded region.


Geometrical Puzzle


Escape to the answer! 


Source 

Geometrical Puzzle - Solution


What was the puzzle?

Let's draw a line from each of vertex to the point at which all 4 regions intersect. This divides the given area into 2 triangles as shown below.

Geometrical Puzzle - Solution

Obviously, here A and A' have equal area as they both share same base QS and height OT. Similarly, the areas of B & B', areas of C & C' and areas of D & D' must be equal. 

Geometrical Puzzle - Solution


Rewriting, A = A', B = B', C = C' and D = D'.



Geometrical Puzzle - Solution
 
Now rewriting respective areas,

Geometrical Puzzle - Solution




It's clear that,

A + B = 32

C + D = 16

Adding above 2 equations gives, 

A + B + C + D = 32 + 16 = 48

But from figure, B + C = 20,

A + 20 + D = 48

A + D = 28.

That's the area of the shaded region which is equal to 28 Sq.cm





Whose Number is Bigger?

Ali and Zoe reach into a bag that they know contains nine lottery balls numbered 1-9. They each take one ball out to keep and they look at it secretly. Then, they make the following statements, in order:

Ali: "I don't know whose number is bigger." 

Zoe: "I don't know whose number is bigger either." 

Ali: "I still don't know whose number is bigger." 

Zoe: "Now I know that my number is bigger!"

Assuming Ali and Zoe are perfectly logical, what is Zoe's smallest possible number?

Whose Number is Bigger?

'This' is that smallest possible number!

My Number is the Bigger One!


First you can read what happened?

Recalling what Ali and Zoe said - 

Ali: "I don't know whose number is bigger." 

Zoe: "I don't know whose number is bigger either." 


Ali: "I still don't know whose number is bigger." 


Zoe: "Now I know that my number is bigger!" 


First statement of Ali indicates that she doesn't have either 1 or 9. If she had 1 (or 9) then she would have an idea that Zoe must have bigger (or smaller) number.

Now Zoe is smart enough to know that Ali doesn't have 1 or 9 which is clear from Ali's first statement. Zoe's first statement indicates that she doesn't have 2 or 8 (& neither 1 or 9). If she had 1/2 (or 8/9) then she could have concluded that Ali has bigger (or smaller) number.

Till now Ali has an idea that the Zoe doesn't have 1,2,8,9. So still Ali can't have 2/8 as in that case too she could have made a different statement. Further if Ali had 3 or 7 (and knowing the fact that Zoe doesn't have 1,2,8,9); Ali could have an idea whose number is bigger as 3 is smallest while 7 is biggest among remaining numbers. That means she doesn't have 3 or 7 ( and 1,2,8,9).

From all the statement Zoe can conclude that Ali doesn't have 1,2,3,7,8,9. In short, Ali must have either 4,5,6. 

Now when Zoe says she has bigger number then it must be either 6 or 7 and Ali having 4 or 5. It can't be 5 as in that case Zoe wouldn't be confident as Ali could have 6.

So the Zoe's smallest possible number is 6.

My Number is the Bigger One!
     

The Unfair Arrangement!

Andy and Bill are traveling when they meet Carl. Andy has 5 loaves of bread and Bill has 3; Carl has none and asks to share theirs, promising to pay them 8 gold pieces when they reach the next town.

They agree and divide the bread equally among them. When they reach the next town, Carl offers 5 gold pieces to Andy and 3 to Bill.

“Excuse me,” says Andy. “That’s not equitable.” He proposes another arrangement, which, on consideration, Bill and Carl agree is correct and fair.

The Unfair Arrangement!

How do they divide the 8 gold pieces?

This is fair arrangement of gold distribution! 

Source 

Correcting The Unfair Arrangement!


How unfair the arrangement was?

First we need to know how 8 loaves (5 of Andy & 3 of Bill) are equally distributes among 3.

If each of them is cut into 2 parts then total 16 loaves would be there which can't be divided equally among 3.

Suppose, each of loaves is divided into 3 parts making total 24 loaves available.

Now, Andy makes 15 pieces of his 5 loaves. He eats 8 and gives the remaining 7 to Carl.

Bill makes 9 pieces of his 3 loaves. He eats 8 and gives the remaining 1 to Carl.

This way, Carl too gets 8 pieces and 8 breads are distributed equally among 3.

Correcting The Unfair Arrangement!
 
Obviously, Carl should pay 7 gold pieces to Andy for his 7 pieces and 1 gold piece to Bill for the only piece offered by Bill. 
 
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