This makes sense to me now. Thanks! Jim On Sun, Apr 10, 2016 at 11:49 PM, Andy Latto <andy.latto@pobox.com> wrote:
On Sun, Apr 10, 2016 at 8:28 PM, James Propp <jamespropp@gmail.com> wrote:
There's still the question of optimality. Last I checked, Andy Latto proposed an argument that I found unconvincing. Were other people convinced? If so, could someone explain to me why you should never jump a frog over a frog or a toad over a toad?
If you jump a frog over a frog, then you have two consecutive frogs, with the space behind them. any Toads that still need to pass these frogs will never be able to, so you had better only do this, if at all, at the very end, after all toads have passes all frogs. Since solutions can be time reversed, reversing the directions that toads and frogs move, the same argument shows that if you jump a frog over a frog, you had better do this only at the very beginning, before any frogs have jumped over any toads. (or to put it another way, if you jump a frog over a frog at the very end, after all the toads have passed over all the frogs, you can only do this from a position that is impossible to reach). So jumping frog over frog, or toad over toad, will always get you stuck.
(I hadn't included this in my argument, because I hadn't realized it was legal to jump a frog over a frog, but now see that it makes no difference whether it's legal).
Andy
Jim Propp
On Sunday, April 10, 2016, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 08/04/2016 16:24, Fred Lunnon wrote:
On Thursday, March 31, 2016, Gareth McCaughan <
gareth.mccaughan@pobox.com>
wrote:
It's discussed very briefly in Winning Ways, which makes the assertion that from the standard initial position you just want to alternate between making as many penny moves as possible and making as many dime moves as possible but says nothing about other configurations or about proofs.
Are you sure this is the same puzzle? That strategy would result in T T - F F -> T - T F F -> - T T F F -> ?! which is plainly a cul-de-sac.
Well, unless I'm misunderstanding (WW is frequently rather terse) they say what I say they say, but of course you're right. I think what they actually mean is: make the obvious single move with one kind of piece, then switch to the other and *from then on* always move as many as possible. So:
T T - F F
single move with T:
T - T F F
as many F moves as possible:
T F T - F T F T F -
as many T moves as possible:
T F - F T - F T F T
as many F moves as possible:
F - T F T F F T - T
as many T moves as possible:
F F - T T
and I think this works in general. But it isn't what they actually say.
-- g
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-- Andy.Latto@pobox.com
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