Friday puzzle -- spiral

David? What happened to Ted?

Don't know how to obfuscate the answer, so I'll hold off.

Remember the woodchipper in Fargo? Ted

I'm thinking ten minutes

No :-(

Okay, Bill, go ahead and post answer

Doesn't look like we he many takers on this one.

Bonus if you can tell what direction any of them are facing when they finally meet ;-)

Re: Okay, Bill, go ahead and post answer

I hadn't thought about the direction of meeting, but I did come up with two answers:

Discrete solution:

Assuming that:

1) each is walking with a 3-foot stride

2) each is re-assessing his/her target location and direction of travel at the beginning

of each stride

Bob, Carol, Ted's replacement and Alice will converge to the four corners of a three foot square in approximately 20 minutes. Then they will travel forever around the three-foot square, always turning 90 degrees after each stride to re-align with their respective targets.

(Of course, if they were to shorten their strides a bit, they could manage to get 'close enough' for practical purposes.)

Continuous solution:

Assuming that each is continuously adjusting his/her direction of travel to

align with his/her target, the foursome will rendezvous in precisely 20 minutes after

walking precisely one mile.

Re: Okay, Bill, go ahead and post answer

And solution method ;-)

Correct, and rationale

Yes, that is correct. I hadn't thought about the discrete case. One interesting thing about the continuous case is that the time and distance is known, but the direction is indeterminate. In fact, they end up coming together spinning infinitely fast!

Dan: the method is as follows:

Consider any point in time. Each of the three will have made exactly the same movement relative to their starting points, so they will always be in a square. That means that at any point in time, Alice is always moving toward Bob, and Bob is moving at right angles to the line to Alice. So there is no component of Bob's motion that affects the distance between Alice and Bob. The rate of closure between Alice and Bob is exactly Alice's speed.

The picture below shows roughly how they would converge. In this drawing, they end up having turned through 270 degrees, and have quickly converged. But if you changed the scale to look at a tiny area near the center, you would find them still in a square, and continuing to converge at this relative rate. So while we know the length of the path has an upper bound of 1 mile, moving along that spiral at a constant speed leads to infinitely fast spinning.

Re: Correct, and rationale

As Alex points out, the closing speed is always the same as the walking speed, but that is only true for the continuous case with theoretically perfect tracking.

In the discrete case, there is a tracking error due to the non-continuous nature of the sampling. This tracking error is negligible to start, but increases as the distance to the target decreases. Eventually, the tracking error goes to 100% (3' @ 3')and the trackers never converge.

This kind of error is inherent in any real-time control system based on discrete sampling (which is almost all of them these days). Even assuming that there is no error in the measurement itself (which is itself a bad assumption), the data is only valid at the time it is taken. Any corrective action will always be 'one step' behind by the time it is applied.

In a case like this, you would typically apply a Kalman filter or other predictive algorithm to better anticipate the target position.