Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2004-08

April 2004

Rendezvous on a Planar Lattice

Steve Alpern and Vic Baston


We analyze the optimal behavior of two players who are lost on a planar surface and who want to meet each other in least expected time. They each know the initial distribution of the other's location, but have no common labeling of points, and so cannot simply go to a location agreed to in advance. They have no compasses, so do not even have a common notion of North. For simplicity, we restrict their motions to the two dimensional integer lattice Z2 (graph paper) and their motions to horizontal and vertical directions, as in the original work of Anderson and Fekete. We mostly retain their assumption that the players start at diagonally opposite corners of one of the unit squares of the lattice, and for this case we obtain a complete list of all optimal strategy pairs for the scenarios with and without a common notion of clockwise.

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