Batch 3 - Class 137 - Crossing the Desert

Pre-Class Problem:

In the Russian Roulette problem above, what if the cylinder is spun after each shot?

Answer: Go first. P(1) + P(2) = 1, and P(1) = 5/6*P(2) [or equivalently, P(2) = 1/6+P(1)]. So P(1) = 5/11 and P(2) = 6/11. Here P(1) denotes the probability of first person dying, and P(2) denotes the probability of second person dying.

Attendance: Muskaan, Aneyaa, Khushi, Arnav, Ahana, Diya, Damini, Siddhant, Zorawar

Class Notes:

Suppose there is a plane which on full tank can fly 1 unit starting from an island (which is the home base). We want to extend the journey of this plane by providing another identical plane which will fly with the plane, and then at some point, transfer fuel mid-air to this plane to get the initial plane go as far as possible. We want the second plane to safely return to the home base after assisting the first plane.

What is the farthest we can get the first plane to go? At what point and how much (distance equivalent) fuel will get transferred?

Answer: Transfer 1/3 tankfull after going 1/3 the unit distance. The initial plane can now go 1+1/3=4/3 units

What if we had two planes to assist the initial plane?

Answer: One of the assist planes goes 1/4 the distance and transfers 1/4 tankfull each to the other two planes. The second assist planes goes another 1/4 distance and transfers 1/4 more to the first plane (So the first plane has got total 1/2 tank full extra). The first plane can now go 1.5 units distance
Note that we are assuming that the first assist plane can not come back, refuel and go back to rescue the second assist plane. Were that to be allowed, the second assist plane could have gone additional 1/3 distance, to increase the range by 1/4+1/3=7/12 i.e. by 0.583

What if we had three planes to assist the initial plane?

Answer: Now the first assist will happen at 0.2 units, second at 0.4 units and the third at 0.6 units. Thus the total range is now 1.6 units

Can you see a general pattern in the above? What is it? Using this approach, and given infinite planes, how far can we get the first plane to? Can we ever double the range?
(MartinShCol 10.9) Now let us assume that this island is not on a flat surface but on earth. The unit range is half the circumference of the earth. We want the first plane to circumnavigate the earth and come back to the island. How many minimum assist planes to we need?

Answer: Two planes. The problem is same as the two assist planes earlier. This makes the first plane go 3/4 around the earth. The two assist planes now go back in the opposite direction to rescue the first plane back, in a fuel transfer scheme similar to what they did in the first stage.

(MartinShCol 10.10) Now, suppose we had a single truck trying to cross a desert, and we are trying to minimize the amount of fuel consumed. The truck can drop off fuel at a certain place, and come back for more fuel. How far can the truck go on 2 truck loads of fuel?

Answer: 1+1/3 (Form a cache at 1/3 distance)

How far can the truck go with 3 truckloads?

Answer: Now the first store will be at 1/5 distance, and the truck will make two trips there to stock 6/5 loads. On the third trip, the truck will have 4/5 units when it reaches the store i.e. total of 2 units. This can take the truck 1+1/3 units further, for a total of 1+1/3+1/5 distance

Can you see a pattern? On distance with truckloads of fuel? on number of stores required?

Answer: The pattern is 1+1/3+1/5+1/7... Number of stores is one less than number of terms.
How far can the truck go? Answer: Since the series diverges, the truck can go an infinite distance.

Homework Problem

What if the truck also wants to come back?
How far can they go on limited fuel loads, and how far can they go if unlimited fuel loads were available?

Answer: The pattern now is 1/2+1/4+1/6+...
How far can the truck go? Answer: Again, the series is divergent and the truck can go any distance.

References:

The Colossal Book of Short Puzzles and Problems, by Martin Gardner