Fat Aunts
January 14th, 2010
Two aunts are living each in her own (0-dimensional) house. There are two non-intersecting (1-dimensional) roads between the houses.
Last year, both aunts were doing a lot of exercise, and so they were slim (0-dimensional). They managed to walk together from House 1 to House 2, taking different roads, while each was holding one end of a rope of length less than L.
This year, they gained weight, and each became a sphere of radius L/2. One aunt is in House 1 and the other is in House 2. Can they exchange houses without bumping into each other (their centers must always remain on the roads)?
January 17th, 2010 at 2:28 am
Solution:
It is impossible for the aunts to exchange houses.
Proof:
Mark route 1 as p:[0,1] to R^n (n is the map’s dimension, 2 in this case)
Mark route 2 as q:[0,1] to R^n (n is the map’s dimension, 2 in this case)
Consider the following function f:[0,1]x[0,1] to R
f(x,y)=dist(p(x),q(y))
f is the distance from each point on route 1 to each point on route 2.
Given the fact that the aunts could walk with the rope, we get that there is a path g:[0,1] to [0,1]x[0,1]
such that:
g[0] = (0,0)
g[1] = (1,1)
g[t] is smaller than L
the aunts could exchange houses, iff there is a path h:[0,1] to [0,1]x[0,1] such that:
h[0] = (1,0)
h[1] = (0,1)
h[t] is greater than L
ofcourse h and g must intersect, therefore it is impossible
January 17th, 2010 at 2:29 am
Trying to sleep, is the ultimate time for solving riddles.
January 17th, 2010 at 2:30 am
and one small correction:
instead of g[t] is smaller and h[t] is greater, i meant to write:
f(g[t]),f(h[t])