Tuesday, April 30, 2019

Trip Around The Earth

Professor Fukano plans to circumnavigate the world in his new airplane. But the plane's fuel tank doesn't hold enough for the trip—in fact, it holds only enough for half the trip. But with the help of two identical support planes (which can refuel him in mid-air) piloted by his assistants Fugari and Orokana, the professor thinks he can make it in one trip. But since all three planes have the same problem of limited fuel, how can they work together to achieve the professor's goal without anyone running out of fuel?

1. The professor's plane must make a single continuous trip around the world without landing or turning around.

2. Each plane can travel exactly 1 degree of longitude in 1 minute for every kiloliter of fuel. Each can hold a maximum of 180 kiloliters of fuel.

3. Any plane can refuel any of the others in mid-air by meeting at the same point and instantly transferring any amount of fuel.

4. Fugari and Orokana's planes can turn around instantaneously without burning fuel.

5. Only one airport is available for any of the planes to land, take off, or refuel.

6. All three planes must survive the experiment, and none may run of fuel in mid-air.


Trip Around The Earth




'This' is how mission is completed!
  

For The Journey Around The Earth


First read T&Cs of the journey! 

Let's assume that the only airport mentioned is located at the top of the earth. 

Recollect all the data given.
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1. The professor's plane must make a single continuous trip around the world without landing or turning around.

2. Each plane can travel exactly 1 degree of longitude in 1 minute for every kiloliter of fuel. Each can hold a maximum of 180 kiloliters of fuel.

3. Any plane can refuel any of the others in mid-air by meeting at the same point and instantly transferring any amount of fuel.

4. Fugari and Orokana's planes can turn around instantaneously without burning fuel.

5. Only one airport is available for any of the planes to land, take off, or refuel.

6. All three planes must survive the experiment, and none may run of fuel in mid-air.

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As per (2), since each plane travel 1 degree of longitude in 1 minute for every kiloliter of fuel, means that plane need 360 minutes (6 hours) and 360 kiloliters of fuel. But remember plane can hold only 180 kiloliters of fuel.

Let's suppose that, all three planes takes off from airport exactly at 12:00 PM towards the WEST.

We will break this 6 hours journey into 8 parts where each plane travels 45 degree of longitude east or west using 45 kiloliters of fuel.  

START :

For The Journey Around The Earth - Start

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PART 1 :

 Exactly at 12:45 PM, all will be at 45 degree angle with reference to the center of the earth.
At this point, each of them will use 45 kiloliters of fuel. Hence, each will have 135 kiloliters of fuel. Here, Orokana gives away 45 kiloliters to each of Fukano & Fugari. So she is left with the 45 kiloliters which she uses to go back to the starting point.

For The Journey Around The Earth - Part 1

 
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PART 2 : 

In next 45 minutes, both Fukano and Fugari moves to 90 degree of longitude spending 45 kiloliters of fuel. Now,here both will have 135 kiloliters each in their fuel tank. Here, Fugari refuels Fukano's fuel tank with 45 kiloliters of fuel; leaving 90 kiloliters in own tank for the backward journey towards the starting point.


For The Journey Around The Earth - Part 2

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PART 3 : 

Fukano travels further, while Fugari is in midway of the backward journey. Again, both spend 45 kiloliters of fuel.


For The Journey Around The Earth - Part 3
 
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PART 4 :

Exactly at 3:00 PM, Fukano reaches at 180 degree while Fugari reaches back to the starting point. Till then, Orokana refuels her plane & takes off towards EAST. She has to take off as Fukano is left with only 90 kiloliters of fuel by which he could travel half of the rest of journey.


For The Journey Around The Earth - Part 4

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PART 5 :

In next 45 minutes, Fukano's plane uses 45 kiloliters further while Orokana travels 1/3rd of Fukano's remaining journey in reverse direction so as to meet Fukano in midway. In process, her plane again uses 45 kiloliters of fuel with 135 kiloliters left.


For The Journey Around The Earth - Part 5

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PART 6 :

At 04:30 PM, Orokana meets Fukano whose plane had no fuel left at the point & refuels his plane with 45 kiloliters of fuel. Remember Orokana's plane consumed 45 kiloliters more till she meets Fukano. Now, since both of them have left only 45 kiloliters in fuel tank, Fugari whose plane standing at airport is refueled at full 180 kiloliters takes of in the direction of EAST.


For The Journey Around The Earth - Part 6


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PART 7 : 

At 05:15 PM, when fuel tank indicators of both Fukano & Orokana are pointing to 0, Fugari meets them & gives 45 (to Fukano) + 45 (to Orokana)  = 90 out of 135 (180 - 45 used since take off) . Now all are left with 45 kiloliters of fuel & 45 degrees of journey is left.


For The Journey Around The Earth - Part 7

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PART 8 : 

And this is how, exactly at 06:00 PM all of them reaches back to the starting point safely.

For The Journey Around The Earth - Part 8


But is this the most efficient way to make trip around the Earth? Certainly not

If plane was built with fuel tank of 360 + then the mission wouldn't have required any assistance. Just because of limited fuel tank, 45 + 45 + 90 + 90 + 90 + 90 + 45 + 45 = 270 Kiloliters of fuel burnt to assist Fukano's plane.



Wednesday, April 24, 2019

Test if You Are a Genius!

People’s Daily,China tweeted out this math puzzle in which each picture represents a number.

The question reads,

"Are you a math genius? Test your brain with this Chinese math puzzle. Did you get it right? (HINT: The correct answer is 16)"

Solve if you can!


Test if You Are a Genius!



Did you figure out how correct answer is 16? 

Did You Prove Yourself Genius?


Look at the question first! 

Let's take a look at the puzzle again.




First thing, we should observe in such puzzles that how objects in the question equation differs from those in given equations.

Here, pair of the sneakers appears as it is in the question while pair of whistles reduced to 1 whistle in question.

And if we look closely the cat, the whistle around the neck is missing in the question.

Now from equation (1), pair of sneakers = 10.

From equation (2), Cat = 5. 

From equation (3), 2 whistles = 4; hence 1 whistle = 2.

Here, we can deduce that cat without whistle in question is CAT - WHISTLE = 5 - 2 = 3.

So the equation in question comes out as, 10 + 3 X 2  = 10 + 6 = 16.

Did You Prove Yourself Genius?


Monday, April 22, 2019

The Green-Eyed Logic Puzzle

In the green-eyed logic puzzle, there is an island of 100 perfectly logical prisoners who have green eyes—but they don't know that. They have been trapped on the island since birth, have never seen a mirror, and have never discussed their eye color.

On the island, green-eyed people are allowed to leave, but only if they go alone, at night, to a guard booth, where the guard will examine eye color and either let the person go (green eyes) or throw them in the volcano (non-green eyes). The people don't know their own eye color; they can never discuss or learn their own eye color; they can only leave at night; and they are given only a single hint when someone from the outside visits the island. That's a tough life!

One day, a visitor comes to the island. The visitor tells the prisoners: "At least one of you has green eyes." 

On the 100th morning after, all the prisoners are gone, all having asked to leave on the night before. 

The Green-Eyed Logic Puzzle

How did they figure it out?


Here is the solution! 

The Green-Eyed Puzzle Solution


Here is that Puzzle! 

Nobody is going to dare to go the guard unless he is absolutely sure that he is green eyed; otherwise it would be suicidal move.

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For a moment, let's assume there are only 2 prisoners named A & B trapped on island.

On first day, A is watching green eyed B & B can see green-eyed A. But both are not sure what is color of their own eyes. Instead, A(or B) waited B(or A) to escape from island since he is green-eyed. Rather both are sure that other too doesn't know anything about color of own eyes.

On next morning both see each other still on island. Here is what A thinks.


If I was non green-eyed then B would have realized that the person pointed by visitor in his statement ('at least one of you have green eyes') is himself. Hence, B would have realized that he is green-eyed & could have escaped easily. Since B didn't try to escape that means I too must have green eyes.

So A can conclude that he too have green eyes. Exactly same way, B concludes that he too has green eyes. Hence, on next day both can escape from the island.
 

Note that, if the night of the day on which visitor made statement is counted then next day would have 1st morning & 2nd night since visitor's visit. Now since A & B left on 2nd night, we can't see anybody on 2nd morning next day.

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Now let's assume there are 3 prisoners named A,B and C trapped on island.

Let's think from A's point of view as an example.What he thinks.

Let me assume I don't have green eyes.Now each B and C could see 1 green-eyed & other non green-eyed person. But still they don't know color of own eyes.So on that night nobody tries to escape.

On first morning I see both B and C still present there.

Now B can think that if he has no green eyes then C could have concluded that the person pointed by visitor's statement ('at least one of you have green eyes') is C himself (as both A & B are non green-eyed. This way, C would have realized that he is green-eyed.

In a very similar way, B would have realized that he too is green-eyed.

Now both of them could have escaped on that night as they are sure that they are green-eyed.

But on the second morning, I see again both of them are still there. So now I can conclude that I too have green eyes.


If A can conclude then why can't B and C?  So after seeing each other on 3rd day, each of 3 can conclude color of eyes is green. Now on 3rd night they all can escape safely.

This is called as inductive logic.

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If observed carefully, 2 prisoners need 2 nights and 3 prisoners need 3 nights to logically deduce that the each of them is green-eyed.

Hence, 100 prisoners would require 100 nights to absolutely make sure that each of them is green eyed.

That's why on the 100th morning next day, there is no prisoner present on island. 


The Green-Eyed Puzzle Solution


Crack 'This' Logic!

IF

1   1   1   1    =  R

2  2   2   2   =  T

3  3   3   3   = E
 
4  4   4   4   = N

Then

 5  5   5   5   =  ?

Crack This Logic!



Skip To The Answer!

Cracked 'THAT' Logic!


Or look at the question first!


 Let's have a look at the given equations once again.

1   1   1   1    =  R
 
2  2   2   2   =  T

3  3   3   3   = E

4  4   4   4   = N

If rewritten after adding numbers & addition is spelled then,

1   1   1   1    =  FOUR
    
2  2   2   2   =  EIGHT

3  3   3   3   = THREE

 4  4   4   4   = SIXTEEN

 Clearly, last letter of spelling is taken.

Hence,


5  5  5  5 = TWENTY

5  5  5  5 = Y  

Cracked 'THAT' Logic!

Tuesday, April 9, 2019

Special Squence Of Numbers

What is special about the following sequence of numbers?

8 5 4 9 1 7 6 10 3 2 0

Special Squence Of Numbers


This is how it's special!

Speciality of Special Sequence


Here is that sequence!

Looking at the special sequence once again.

8 5 4 9 1 7 6 10 3 2 0

If they are spelled as in sequence -

Eight   Five   Four   Nine   One   Seven   Six   Ten   Three   Two   Zero 

Yes, you got it right. They are in an alphabetical order.


Speciality of Special Sequence

Monday, April 8, 2019

The Hiker's Dilemma

A hiker comes across an intersection where three roads cross. He looks for the sign indicating the direction to his destination city. He finds that the pole carrying three city names and arrows pointing to them has fallen. He picks it up, considers it, and pops it back into place, pointing out the correct direction for his destination. How did he do it?

The Hiker's Dilemma


This is how he did it!


Hiker Got The Right Direction


What was the dilemma? 

Since he knew very well from which city he came; he just oriented that arrow in right direction. And obviously hence rest of all arrows are pointed in the right directions as well. This is how he found his right way!

Hiker Got The Right Direction

Tuesday, April 2, 2019

A Fractional Pink Shade

What fraction of this figure is shaded with the pink color?

Find Area Of A Fractional Pink Shade

Get the answer here!


Author : Ed Southall of Solve My Maths

Area Of A Fractional Pink Shade


A look at the question first!

Let's first recall the formula for the calculation of area of a triangle.

Area of triangle =  1/2 x Base x Height

Let's assume the side of the square is 1.

Now the triangle with the pink shade & triangle opposite to it are similar triangles. Similar triangles are triangle whose sides are in proportion with each other.

Since here base of pink triangle is double of un shaded opposite triangle, the height of pink triangle must be double of that smaller triangle.

But together, heights of both triangles must be equal to side of the square i.e. 1.

And hence, height of smaller triangle must be 1/3 & that of pink 2/3. (h + 2h = 3 ; h = 1/3).

So,

Area of Pink Triangle = 1/2 x Base x Height = 1/2 x 1 x 2/3 = 1/3.

So the area of pink shaded part is 1/3rd of total area occupied by square.