Lion and Man
Lion and Man
A man and a lion are inside a disk of radius 1 with center . Both have a top speed of 1. Can the man choose a strategy to avoid being captured by the lion?
S
To use the Demonstration, alternate the man's move (button) and the lion's move (choose a point inside the small circle and Alt+click).
A solution consists of a polygonal line …… of length , with , where ==S. The lion can run a path …… of the same length and can choose the strategy so that the distance converges to 0, but only as time goes to infinity.
M
1
M
2
M
k
M
k+1
s++⋯++⋯
1
2
1
3
1
k+1
s=1-
s
1
s
1
M
1
M
1
L
1
L
2
L
k
L
k+1
M
k
L
k
The man's strategy is as follows. The point is constructed toward the center of the disk so that the segment is of length and perpendicular to . The square of the distance of to the center of the disk is ≤+≤+++⋯+<+++⋯+=+1-<+<=1.
M
k+1
M
k
M
k+1
s
k+1
M
k
L
k
M
k+1
S
2
S
M
k+1
2
S
M
k
2
s
k+1
2
s
1
2
s
1
2
2
1
2
3
1
2
(k+1)
2
s
1
2
s
1
1·2
1
2·3
1
k·(k+1)
2
s
1
2
s
1
k+1
2
s
1
2
s
2
(+s)
s
1
So the man is always inside the disk (explanation 1 for ).
k=1
The distance between the man and the lion is (explanation 2 for ).
≥+->0
M
k+1
L
k+1
2
M
k
L
k
2
s
k+1
s
k+1
k=1