[extropy-chat] Spike's Aeroplanes Puzzle

[Hal Finney]

Ben -

Typing on my wireless, so I'll be concise.

I can improve yr 3 plane soln. Let the 1st plane transfer r/3 its fuel 
to the 2nd plane. Then the 2 planes fly an additional r/9. The 2nd plane 
transfers 4r/9 to the final plane. This fills the final plane & leaves 
the 2nd plane w/ 4r/9 in its tanks so it can fly home. This gives a 
range of 13r/9.

But for a globe we can do better. Let the 2nd helper plane fly the 
opposite direction. The 1st helper transfers r/3 as usual. This gives 
the main plane a range of 4r/3. The 2nd plane flies r/3, meets the 1st 
as it is running out, xfers r/3, and both fly back. This gives a range 
of 5r/3, so r = 0.6 of the circumference.

I was not sure how to generalize to N planes for the circular case, 
although for a straight line I think the increment goes down by a factor 
of 3 for each added plane.

Hal

On Sat, 13 May 2006 3:19, ben wrote:
> 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
> _______________________________________________
> extropy-chat mailing list
> extropy-chat at lists.extropy.org
> http://lists.extropy.org/mailman/listinfo.cgi/extropy-chat