"Go The Distance"

There are 50 bikes with a tank that has the capacity to go 100 km. Using these 50 bikes, what is the maximum distance that you can go? 

Here is the maximum distance calculation!

Maximizing The Distance!

What was the challenge?
 
Remember, there are 50 bikes, each with a tank that has the capacity to go 100 kms. 

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SOLUTION 1 : 

Any body can think that these 50 bikes together can travel 50 x 100 = 5000 km. But this is not true in the case as all bikes will be starting from the same point. And we need to find how far we can we go from that point. 

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SOLUTION 2 : 

Just launch all 50 bikes altogether from some starting point and go the distance of only 100 km with tanks of all bikes empty in the end.

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SOLUTION 3 : 

1. Take all 50 bikes to 50 km so that tank of each is at half.

2. Pour fuels of 25 bikes (half filled) into other 25 bikes so that their tanks are full.

3. Now, move these 25 bikes to another 50 km so that again their tanks are at half.

4. Pour fuel of 12 bikes into other 12 so that we have 12 bikes with full fuel tank. Leave 1 bike with half filled fuel tank and repeat above.

So for every 50 km distance, half of bikes are eliminated as - 

50 ---> 25 ---> 12 ---> 6 ---> 3 ---> 1

The last bike left with it's tank full can go 100 km. So. the total distance that can be traveled in the case is 5 x 50 + 100 = 350 km. 

However, we have wasted 1/2 fuel each whenever odd number of bikes are left i.e. at 25 and at 3. 

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SOLUTION 4 :

Let's optimize little further so that the 1/2 fuel is not wasted whenever odd bikes are left.


1. Take all 50 bikes to 50 km so that tank of each is at half.

2. Pour fuels of 25 bikes (half filled) into other 25 bikes so that their tanks are full.

3. Now take these 25 bikes to another 20 km using 1/5th (20/100) fuel of each. 

4. Make 5 groups of 5 bikes each. From each group, use 4/5th fuel of 1 bike to fill tank 1/5th emptied tanks of other 4 bikes.

5. Leave bike with empty tank and take 20 bikes to next 50 km. And again after 50 km, pour fuel of 10 bikes into other 10 to eliminate 10.

6. After moving 10 bike for another 50 km, again pour fuel of 5 bikes into another 5.

7. Now take these 5 bikes to another 20 km using 1/5th (20/100) fuel of each.

8. Use 4/5th fuel of 1 bike to fill tank 1/5th emptied tanks of other 4 bikes. 

9. Now these 4 bikes again taken to another 50 km where 2 more are eliminated by taking half of their fuel to fill tanks of other 2.

10. After taking those 2 bikes for another 50 km distance, 1 can be eliminated by taking away it's half fuel to fill up the tank of other bike.

11. The last bike can now go another 100 km distance as it's tanks is full.

To summarize,

50 ---50km---> 25 ---20km---> 20 ---50km---> 10 ---50km--- > 5 ---20km--- > 4 ---50km ---> 

--->2 ---50km---> 1 ---100km ---||

Total distance that can be traveled = 5 x 50 + 2 x 20 + 100 = 390 km.  

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SOLUTION 5 : 

Now we have got the idea from SOLUTION 4 how to maximize the distance further.

Instead of waiting for tanks to be at half or 4/5th we should empty the tank of 1 bike into others at the point where that bike has sufficient fuel for this process.

For example, to have 49/50th fuel in tank of 1 bike at some point, all bikes need to be taken so that 1/50th of each is used up. Since the bike goes 100 km with full tank, with 1/50th fuel it can go 100 x 1/50 = 2km distance.

In short, after 2km distance 49/50th fuel of 1 bike can be used to fill 1/50th empty tanks of other 49 bikes. Now, that 1 bike with empty tank can be left there.

For next phase, we have, 49 bikes. Now, after using up another 1/49th fuel for another distance of (1/49) x 100 = 100/49 km, the 48/49th fuel left in any one bike can fill up the tanks of other 48 bikes (each with 1/49th part is empty). Then, these 48 bikes can be taken for the next phase.

Now, again after consuming 1/48 fuel for another distance of 100/48km, 47/48th of fuel from 1 bike can be used to fill tanks of other 47 bikes (each bike with 1/48th tank empty after traveling 100/48km). So, now 47 bikes can be taken for the next phase.

This way, we are making sure that at each phase 1 bike uses it's all fuel to make tanks of other full.

Repeating this process, till 1 bike left which can go further 100km with full tank.

So the total distance that can be covered is - 

100/50 + 100/49 + 100/48 +.................100/1 = 449.92 km.

And this is the maximum distance that we can go with 50 bikes.


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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.

'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 :

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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.

 
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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.

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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.

 
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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.

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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.

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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.

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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.

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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.

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.