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        riddles >> medium >> Sir George's Land Grant
        
(Message started by: TimMann on Oct 24^th, 2003, 11:33pm)

Title: Sir George's Land Grant
Post by TimMann on Oct 24^th, 2003, 11:33pm

Here's a puzzle that I cribbed from the online book celebrating G4G1 (the Gathering for [Martin] Gardner #1), available at http://www.g4g4.com/contentsmmpp.html.

"To reward you for killing the dragon," the Queen said to Sir George, "I grant you the land you can walk around in a day." She pointed to a pile of wooden stakes. "Take some of these stakes with you," she continued. "Pound them into the ground along your way, and be back at your starting point in 24 hours. All the land in the convex hull of your stakes will be yours." (The Queen had read a little mathematics.) Assume that it takes Sir George 1 minute to pound a stake and that he walks at a constant speed between stakes. How many stakes should he take with him to get as much land as possible?

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Edit: changed link to point to a legitimate copy of the book.

Title: Re: Sir George's Land Grant
Post by towr on Oct 25^th, 2003, 9:55am

::[hide]since the world is round, one stake could claim the world.. Though the queen would probably not (want to) see it that way..[/hide]::

Title: Re: Sir George's Land Grant
Post by TimMann on Oct 25^th, 2003, 11:09am

Hee hee, towr. No need to hide that answer.

Sir George lived long before Columbus, so he and the queen were assuming a flat earth. This should be obvious, as convex hulls aren't well-defined on a sphere. ;D

Title: Re: Sir George's Land Grant
Post by Barukh on Oct 26^th, 2003, 12:29am

Here's my attempt: [smiley=blacksquare.gif][hide]17 stakes. It's the number that maximizes (1440-n)²*cot([pi]/n)/n[/hide][smiley=blacksquare.gif]

Title: Re: Sir George's Land Grant
Post by towr on Oct 26^th, 2003, 9:08am

[edit]I can't seem to get it right..

Anyway, I finally get the same answer as Barukh so forget whatever I wrote here before :P

Well, a slight difference, I maximized [hide]v²·(n - 1440)²·cot(pi/n)/(4·n)[/hide], but that doesn't matter for n, just for the final area[/edit]