[extropy-chat] Spike's Aeroplanes Puzzle

[Kevin Armstrong]

Hal, you have the right idea for the linear solution, though make sure you
conclude by saying 13r/9 = W, so that r = 9/13 W.

In general, for n planes, I believe we need r + 1/3 r + 1/9 r + ... +
(1/3)^(n-1) * r = W.  If memory of geometric series serves me, for n
approaching infinity, we get the left hand side equal to 1/(1 - 1/3) r = 3/2
r, so infinitely many planes would require r = 2/3 W.  Not exactly an
overwhelming improvement over the two plane solution, or even over the one
plane solution.

-Kevin


On 5/13/06, Hal Finney <hal at finney.org> wrote:
>
> 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
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