Re: Protecting a point (math puzzle)

"A.B." <ab2@xxxxxxxxxxxxxxx> writes:

I thought of the following puzzle, I'm not sure of the answer...

There's a game in a space of N dimensions. The merchant is a point M
in this space. There are hidden thieves in this space, and the
merchant wants to protect himself by hiring swordsmen, and placing
them around him at S1,S2...Sp. The merchant is protected from the
thieves, if and only if for any T in R^N,
d(T,K) > d(T,{S1...Sp}). What is the minimum amount p* of swordsmen
the merchant needs to hire ?

For N=2 and N=3 I find a square and a cube, p*=4 and p*=8... it seems
to indicate p* = 2^N but it may not be so. Any ideas ?

As others have said, this doesn't make sense because you could have T=K.
But suppose you have a nonzero lower bound r on the distance of the nearest
thief, so the requirement is d(T,K) > d(T,{S1,...,Sp}) for all T in R^n with
d(T,K) > r. Then p* = N+1: you can put the swordsmen at the vertices of a
sufficiently small N-simplex with barycentre at K. On the other hand,
you can't get by with N, because these would be on some hyperplane, and
a thief on the line through K normal to that hyperplane (on the same side
as K, if K is on one side) will be closer to K than to the swordsmen.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

Relevant Pages

  • Re: Protecting a point (math puzzle)
    ... There are hidden thieves in this space, ... merchant wants to protect himself by hiring swordsmen, and placing ... The merchant is protected from the ...
    (sci.math)
  • Re: Protecting a point (math puzzle)
    ... There's a game in a space of N dimensions. ... The merchant is a point M ... There are hidden thieves in this space, ... merchant wants to protect himself by hiring swordsmen, ...
    (sci.math)
  • Protecting a point (math puzzle)
    ... There's a game in a space of N dimensions. ... The merchant is a point M ... There are hidden thieves in this space, ... merchant wants to protect himself by hiring swordsmen, ...
    (sci.math)