Bullet Richocheting in a Room

lugita15

Here is an interesting puzzle from Steve Landsburg's excellent blog:
http://www.thebigquestions.com/2012/05/22/tuesday-puzzle/

"You’re in a rectangular room. Elsewhere in the room is a man with a gun, who shoots a bullet in a random direction. The bullet careens around the room, bouncing off walls, until it hits either you or one of the various punching bags you’ve placed around the room for purposes of absorbing the bullet. The punching bags must be positioned before you know the random direction of the bullet (though you do know both your own location and the bad guy’s location, neither of which you can change). How many punching bags do you need to guarantee your survival?

This being a math problem, you should treat the room as two dimensional, and yourself, the bullet and the punching bags as points."

Whovian

None. As you're a point, and so is the bullet, the probability that, when shot in a random direction, the bullet hits you, for any finite amount of time, is 0.

lugita15

Of course, the problem is asking you to survive with certainty, not just with 100% probability.

collinsmark

From the comments section of the website having the original riddle:

That was from the author of the riddle. So apparently a probability approaching 100% isn't good enough for the answer. It needs to be 100% guaranteed.

collinsmark

My proposed answer:

lugita15

Even a probability equalling 100% is not good enough. Certainty is required.

lugita15

Both the shooter and the punching bag are points, so it will just be as if the bullet is leaving the punching bag.

Topher925

Zero. Point the shooter perpendicular to the wall so the bullet bounces back and hits him.

Whovian

But the problem did say he shot in a random direction.

Topher925

The direction can be random. It just needs to be perpendicular to a wall.

Whovian

So you know the direction beforehand, then?

And I don't understand how an angle of, say 30˚ relative to a random wall in a rectangular room could be perpendicular to the wall.

Topher925

No, you don't know the direction the bullet will travel so its still random. The room is rectangular, so 4 sides. There is a random probability of 25% that the bullet will hit one of the 4 walls. Just don't stand between the wall and the bullet and you'll survive as long as the shooter is standing perpendicular to the wall.

Whovian

But what if it ricochets, bouncing off one wall, so it hits another wall, and ricochets back towards you? We were never told the bullet's direction would be perpendicular to the wall, and the only situation in which it would just bounce back and hit the shooter is if it's perpendicular to the wall it hits.

lugita15

It doesn't matter what direction the shooter is faced in, regardless he will still shoot at a random angle, so the probability that he will fire perpendicular to a wall is zero.

Borek

No matter where I put the bag and myself, if we are points and we are not in the same place, it is always possible that the guy with the gun will stay between me and the bag and will aim exactly at me. So I don't see how it can be solved in a general case. It is not clear if putting bag and myself in the same position makes me safe or not.

collinsmark

The riddle statement specifically said, "This being a math problem, you should treat the room as two dimensional, and yourself, the bullet and the punching bags as points."

What that implies is that the gun shooter and bullet (before the bullet is shot, if it's not already inside a bag) can be treated as a single point. The bullet stops the moment the bullet location is exactly equal to to the location of a bag or you. Placing a punching bag at the location of the shooter satisfies the mathematical requirement for stopping the bullet.

Think of it as placing the shooter, gun, and bullet inside a single punching bag, if that helps.

[Edit: or alternately, think of it as sticking a single punching bag inside the bullet (which happens to be at the exact location as the shooter and gun). In either case, the bullet and bag intercept, from a mathematical perspective. The requirement is satisfied. The bullet is stopped.]

lugita15

Remember, your location and his location are fixed, and you can put the punching bags wherever you want. If you want to prevent him from shooting directly at you, just put a bag between you and him.