[extropy-chat] Spike's Aeroplanes Puzzle

[Hal Finney]

On Sat, 13 May 2006 21:42, spike wrote:
> Guys, keep working.  I don't see the right answer yet, or rather I 
> found a
> general solution that works.  Then I realized I did not account for 
> planes
> going around the other direction.  {8-]  spike

Okay, I tool another look @ this, for the 3 plane case.

What I realized is that it's best to transfer fuel as early as possible. 
Ideally with 2 planes the 1st plane would xfer all its fuel immediately. 
We can't do that because the 2nd plane can't hold it, but if we could it 
would double the range.

So what we should do with 3 planes is to transfer as soon as the other 
planes can hold it. This is after r/4. We can transfer 1/4 tank to each 
of the other 2 planes, filling them up, and still make it back.

Then they fly another r/4 and the 2nd plane transfers 1/4 tank to the 
final one. The 3rd plane has had a net addition of 1/2 tank and can fly 
3r/2, better than my 13r/9 solution (but not as good as my 5r/3 for 
going around the world).

The N plane solution will use the same idea, each plane turns back as 
soon as the others can hold its transfered fuel. Just as with N=3 we 
were able to transfer 1/4 tank twice, for N it turns out we can transfer 
1/(N+1) tank, (N-1) times. This gives a total range of r * (1 + 
(N-1)/(N+1)). I guess the real question is whether there is a proof of 
optimality!

Hal