[extropy-chat] Spike's Aeroplanes Puzzle

[ben]

OK, it's been almost a month now, and no-one's bitten.

I'm curious to know the answer to this. Obviously it involves calculus,
and that's beyond my current capability to get my head around, but
here's my naive take on the problem.


Spike riddled:

> For a plane to fly around the world without landing, its tank would 
> need to hold sufficient fuel to go all the way around.  But what if 
> you had two identical planes, with fuel transfer capability.  They 
> could take off together, fly some distance, one transfers a quantity 
> of fuel into the other plane and immediately turns back, returning to
> the point of origin. The other plane, which received the fuel, flies
> on around.
> 
> 1.  What is the necessary minimum range of the two planes such that 
> the two could fly a ways, do a transfer, one plane turn around and go
>  back to the start and the other go around?


Ben squeezed hard on his tiny brain, and came up with:

W = dist round the world
r = range of 1 tank of fuel
Maximum amount of fuel that can be transferred = r/3
(If it was any other amount, then either the plane won't make it back,
or will be back with fuel to spare, so fuel transferred must be r/3 for
max. effect).

So if plane 2 can make it round the world with 1 and 1/3 tanks of fuel,
W = r + r/3
Er, my algebra is still a bit dodgy, but i think that's 0.75W. So the
planes have a range of 3/4 the distance round the world.

> 2.  What is the necessary minimum range capability if one had three 
> such planes?

The same logic applies to each individual plane, i.e., maximum fuel
donation will be r/3, so with 2 donor planes, we have 2r/3 fuel
available at point r/3.
But the single plane that continues can't accept more fuel than it has
used so far (r/3), so one of the planes has to donate 1/6 it's fuel to
each of the two other planes, which would add 1/12r to their journey.
But then one would later transfer 1/3 of it's extra fuel to the
remaining plane, so that it had enough to get back, which would add
another r/18. The final plane would use r + r/3 + r/18 to get round the
world. r = 0.72W



> 3.  What is the necessary range capability if one has N planes? (This
> one is cool).


With two planes, r = 0.75W, with 3 planes, r = 0.72W, and every extra
plane will remove a smaller distance from the total - transfer r/3 to 
N-1 planes, and each one in turn transfers 1/3 of that, etc. They all 
have to get back, from further and further away, which needs more and 
more fuel, so a smaller and smaller proportion of the original fuel is
avaliable for the final plane.

I don't know enough maths to cope with this, it's obviously calculus or
something, but it's going to be an asymptote. I think.

So N planes will each have a range of somewhere between 0.75W and 0. I
have a feeling that an infinite number of planes would need a range of 0?

Unless i've got myself horribly confused (not difficult).

But i don't know how to tell the range with N planes.

Do tell, uncle Spike, please?

(It's odd that problems involving statistics have people feverishly
pounding their keyboards, but this one hasn't drawn a single post.
Unless it's because aeroplanes are boring, whereas zorfs and envelopes 
are fascinating).

ben