Celebrity Problem Celebrity Invanessa Marriot is being…

     Celebrity Problem
     Celebrity Invanessa Marriot is being chased by the notorious paparazzi Gra-
cias Prego. She has shrunk herself to a point x in the unit square [0, 1]2 . Not to
be out-done, Gracias has shrunk himself to a point y = x in the same square.
He has the latest laser camera and would love to get a picture of Invanessa in
her reduced state. He points his camera and it emits a zero thickness light ray.
If it hits a wall it will be reflected. He wants to shoot the ray so that it reaches
Invanessa. She however, can place guards that are points Z1 , Z2 , . . . , in the
square and if the beam tries to go through one of the guards it is blocked. How
many guards does Invanessa need so that no beam from Gracias can reach her?
Will a finite number do?
     Solution: The following solution was provide by Karthik Lakshmanan.
     The first thing we do is remove the complexity due to reflections. We consider
[0, 1]2 to be just one square in a partition of the plane into unit squares. If
two squares share an edge then one is the reflection of the other in that edge
(see diagram). A light ray from Gracias is now just a line. When it meets a
wall, instead of reflecting, it moves into the reflected copy of the square that is
adjacent to it. Thus instead of one Invanessa to aim at, Gracias has a countable
number to aim at. On the other hand, each guard can protect a countable
number of Invanessa's.
     In the diagram, the green mark represent the actual location of Invanessa
and the red marks are the "virtual" copies. If the co-ordinates of Invanessa
are (a, b) then Gracias needs to choose one of the targets (2m ± a, 2n ± b) for
m, n  Z = {0, ±1, ±2, . . . , }.
     If the co-ordinates of Gracias are (c, d) then this means that the laser beams
must be one of the lines:

                   (y - d)(2m ± a - c) = (x - c)(2n ± b - d).

The points
                    (x, y) = (m ± a/2 + c/2, n ± b/2 + d/2)                     (1)
lie on these lines.
    It follows that placing guards at the 16 points

                        (a/2 ± c/2, b/2 ± d/2)
                        (a/2 ± c/2, 1 - (b/2 ± d/2))
                        (1 - (a/2 ± c/2), b/2 ± d/2)
                        (1 - (a/2 ± c/2), 1 - (b/2 ± d/2))

will protect Invanessa from Gracias.
    Here we have assumed that a < c, if not interchange a and c. Similarly for
b, d.
    Acknowledgements: A special thanks to Karthik. Thanks also to the
following people who corresponded with us on the problem: Vahram Avagyan,
Raghu Chaitanya Doppalapudi, Abhay S Harpale, Gustavo Lacerda, Mircea


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Alexandru Petrache, Vibhu Sharma, Yuanming Yu.
Thanks also to Peter Winkler for telling us the problem.




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