[Peekay]

You are at the center of a unit circle (radius=1). A hungry lion is on the circumference and for reasons unknown can only run on the edge of the circle. He is, however, twice as fast as you are and his reflexes are instant (response time=0). So, in whichever direction you start to run, he will instantly run towards that spot on the circle that you are most likely to cross. Is there any way you can escape? (To keep things simple, please consider the lion and yourself as point entities!)

[Guest]

Trace a tangent to the circumference in point L, which is where the Lion stands. Trace a perpendicular line to such tangent at point L. the best option to escape the Lion is running in the perpendicular line's direction.

[k-man]

If you run in a straight line you will not escape

You should continuosly run in the direction opposite the direction to the lion. In other words, your back should always face the lion. As you start moving in the direction opposite the lion, the lion starts moving around the circle (he has to pick either side). You then adjust your course, so that you are moving away from the lion. Your trajectory will be a spiral.

[Guest]

You move a tiny bit off of the center, and the lion moves to intercept. When he is in place, you run in a straight line in the opposite direction. You travel r; the lion must go halfway around the circle, pi*r. Even traveling at twice your speed, he cannot catch you, traveling more than 3 times the distance.

[Guest]

You are at the center of a unit circle (radius=1). A hungry lion is on the circumference and for reasons unknown can only run on the edge of the circle. He is, however, twice as fast as you are and his reflexes are instant (response time=0). So, in whichever direction you start to run, he will instantly run towards that spot on the circle that you are most likely to cross. Is there any way you can escape? (To keep things simple, please consider the lion and yourself as point entities!)

if u run towards the point that is opposite to the position of the lion, the lion will not have enough time to reach there when u reach it.

lets say your speed is v, the speed of the lion is 2v

the time it will take u to reach that point is distance/speed = 1/v

the distance the lion needs to travel is half the circumference which is: 2(pi)r / 2 = (pi)r = pi(1) = 3.141...

the time the lion needs to get there is 3.141/(2v)

which is greater than 1/v for any v..

maybe i understood the question wrong, or the wording is not accurate.

[Peekay]

My mistake. The lion's speed is 4 times yours - not twice, as originally stated. Sorry for the confusion!

[Guest]

You are at the center of a unit circle (radius=1). A hungry lion is on the circumference and for reasons unknown can only run on the edge of the circle. He is, however, twice as fast as you are and his reflexes are instant (response time=0). So, in whichever direction you start to run, he will instantly run towards that spot on the circle that you are most likely to cross. Is there any way you can escape? (To keep things simple, please consider the lion and yourself as point entities!)

[Guest]

Run .4 units from the center, then run in a circle at .4 units from the center until you are on the opposite side of the lion. Once on the opposite side all you need to run out is .6 units, while the lion runs 3.14 units, which is greater than the 2.4 units he can run in that time.

[Thalia]

1. Use your cell phone that you would NEVER go anywhere without and call animal control.

2. Hope that the ground is soft and start digging.

3. Wait until the lion gives up or falls asleep. Maybe throw your shoe as a distraction.

[Guest]

run in spiral:)

[Guest]

A revised version of Imhere's solution, run 0.5 units (half r) from the centre. The lion will now be on the side of the circle .5 units away. Turn 180degrees and begin to move at 0.1*your max speed. This means you travel 0.1 unit the lion travels 4. Reach 0.4 units from centre, turn 180 degrees and run at full speed out of the circle in the direction you travelled to start with. The lion will be on the other side of the circle and will not be able to cover the distance of pi units in the time it takes you to run 0.6 units.

[Guest]

I think the important part is that you are inside a circle. Simply give the lion a piece of pi and while he's eating it, you can make your escape.

[Guest]

you run diametrically away from the lion till say 0.99 (short of 1); by then the lion will have run 3.96 along the circle, that is 0.8184 (3.96-pi) beyond your expected point of interception with the circle; now safely cross the circle's line by cocking a snook at the lion

[Peekay]

You guys are awesome!!! I really enjoyed the exchange of ideas including the cocky ones.

The 2 key factors here are:

1. The lion is 4 times faster than you

2. He anticipates your every move with a zero response time, which means he will be at the spot you are most likely to cross the circle

Also, do remember that while the lion can run 4 times faster than you, he doesn’t necessarily have to. He will always try to anticipate your move.

Regarding k-man’s solution (#3):

While the spiral trajectory seems the most elegant mathematical solution, I am not too sure that it would work with the 2 points stated above. I think (don’t have a formal proof yet), as you approach the edge of the circle, the lion would be able to keep up with you with his faster speed. K-man, please correct me if my logic is in error.

Solution:

If you run along a circle of radius 0.25, the lion would be just able to keep up with you (at 4 times the speed). If you are slightly inside that circle (radius just less than 0.25), you will be able to gain on the lion. You keep running along that circle until the lion is diametrically opposite to you. Then head straight for the edge. You will have to run slightly more than 0.75 units while the lion has to cover pi!

Imhere (#8) was on the right track, but his numbers were a bit off!

[Guest]

wow......ummm im thinking keep yourback to him n run in his direction to confuse him. therefore making him face you which is on the other side

[Guest]

You are at the center of a unit circle (radius=1). A hungry lion is on the circumference and for reasons unknown can only run on the edge of the circle. He is, however, twice as fast as you are and his reflexes are instant (response time=0). So, in whichever direction you start to run, he will instantly run towards that spot on the circle that you are most likely to cross. Is there any way you can escape? (To keep things simple, please consider the lion and yourself as point entities!)

As a literalist, riddles like this amuse me. The answer is quite clearly "Yes".

[Guest]

the lion to the most far point is the opposite point. The time to that point is 3.14/4.

If the person want to escape the lion, he must close the circumference as near as 3.14/4=0.785.

Create a circle with radium (1-3.14/4) = 0.21499999999999997.

The total time of lion to rotate full circle is 2*3.14/4=1.57

The total time of person rotate the created circle is 2*3.14*0.21499999999999997/1=1.3501999999999998.

This means the angular velocity of person on created circle is grater than the angular velocity of lion.

So the escape strategy is

1. run to point 1-3.14/4.

2. run along the circle with radium (1-3.14/4)

3. run and run, until the lion and person on the most far distance

4. move straight to the edge and escape

You are at the center of a unit circle (radius=1). A hungry lion is on the circumference and for reasons unknown can only run on the edge of the circle. He is, however, twice as fast as you are and his reflexes are instant (response time=0). So, in whichever direction you start to run, he will instantly run towards that spot on the circle that you are most likely to cross. Is there any way you can escape? (To keep things simple, please consider the lion and yourself as point entities!)