Lion and Man

lion's initial position

man's initial position

man's move

lion's move: Set the lion's new position

L

k + 1

with Alt+click inside the small disk.

reset

explanation 1

explanation 2

Explanations can be shown after the man's first move and before the lion's first move.

A man and a lion are inside a disk of radius 1 with center

S

. Both have a top speed of 1. Can the man choose a strategy to avoid being captured by the lion?

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

M

1

M

2

…

M

k

M

k+1

…

of length

s

1

2

+

1

3

+⋯+

1

k+1

+⋯

, with

s=1-

s

1

, where

s

1

=

M

1

=

M

1

S

. The lion can run a path

L

1

L

2

…

L

k

L

k+1

…

of the same length and can choose the strategy so that the distance



M

k

L

k



converges to 0, but only as time goes to infinity.

The man's strategy is as follows. The point

M

k+1

is constructed toward the center of the disk so that the segment

M

k

M

k+1

is of length

s

k+1

and perpendicular to

M

k

L

k

. The square of the distance of

M

k+1

to the center

S

of the disk is

2



M

k+1

S

≤

2



M

k

S

+

2

s

k+1

≤

2

s

1

+

2

s

1

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-

1

k+1

<

2

s

1

+

2

s

<

2

(

s

1

+s)

=1.

So the man is always inside the disk (explanation 1 for

k=1

).

The distance between the man and the lion is



M

k+1

L

k+1

≥

2



M

k

L

k



+

2

s

k+1

-

s

k+1

>0

(explanation 2 for

k=1

).