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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
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| ``Re: Exciting Problem''
by rspuzio on 2008-07-07 23:57:20 |
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| There is a standard trick for solving all problems of this sort.
Reflect the triangle to obtain a lattice. Any path will then
become a straight line segment and the number of bounces will become
the number of times this segment intersects the lattice. For
simplicity, set up an oblique coordinate system in which a
the point C is the origin and in which A has coordinates (1,0) and
B has coordinates (0,1). Then points of the lattice are points
with integer coordinates, and a line is part of the lattice if and
only if it is of the form x = n, y = n, or x+y = n, where n is an
integer. Reflections of C are points such that x is congruent to
y modulo 3, reflections of A have x congruent to y+1 modulo 3 and
reflections of B have x congruent to y+2 modulo 3.
Then a line from (0.0) to (m,n) will intersect n-1 lines of the form
x=integer, intersect m-1 lines of the form y=integer. and intersect
m+n-1 lines of the form x+y = integer. Hence, there will be a total
of 2m + 2n - 3 bounces. Now your problem reduces to counting
solutions of a simple linear equation: Given an integer b, how
many pairs (x,y) are there suc that x is congruent to y modulo 3
and 2m + 2n = b + 3ãï¼ |
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