bonanova

Define "going slower." What does it mean? Longer elapsed time before reaching South Pole We could find an arbitrarily long path and define speed = (distance between poles)/(elapsed time.) This definition gives one value of speed for the entire trip and does not depend on the layout of the path. We could go slower under this definition by flying endlessly at night at a constant latitude. Lower path speed. This means speed = ds/dt where s is path length. Here speed is an instantaneous value. With this definition we could go slower just by stopping to rest for a while. That is, we could make our (instantaneous) speed zero. That is very slow. Also define "best path." I think of best path as most efficient - shortest path or earliest arrival. Going slower seems to ask for the path that will tire him the least. So, instantaneous speed, average speed, longest transit time, shortest path, rest stops for the reindeer to regain strength, least time between houses so children don't have to wait for their toys, lowest to the ground so everyone can hear HO HO HO clearly ... there are a lot of things that might make a path "best."

Your second answer is the one I was looking for. It's a result that was interesting to me.

Sounds like an OK approach. You would need to do some (messy) integrals to account for the points being randomly chosen. I like to think of this as more of a logic puzzle than a math exercise. Is there a simpler way to think about the problem?

It's hard to envision Santa being a tachyon at all, much less a slow one. And I'm not sure of the definition of best.

Your conjecture of 0.02 probability of reaching parity of heads and tails for a coin with p(H) = .99 is correct. A biased coin with probability pmin of the less-likely outcome behaves, with probability 2pmin, like an unbiased coin. I'm not sure which case you're unconvinced of, (0, 1) or [0. 1]. Since the closed case seems clear, here's an expanded description of the open case:

Are the four expected values equally spaced?

That explains it, kink of like a rolling action. Thanks.

Very nice. I was stuck at six mirrors, for reasons that now are obvious: Two lights for free, then a mirror each for the last six.

Hi santhu, welcome to the Den. Your answer is close but not quite it.

I have thought about this, and I conclude that I don't get it. I'm seeing this as an inked stamp. Wouldn't you have to replace some spacers with inked ones? Can you touch the paper twice ... ?

It must be concave?

The coveted boanova gold star for creativity. I think you even threw in a great-grandfather and a great-grandson for good measure.

After watching "I Robot," I want these to be good and evil robots.

I searched up to 1020and didn't find any more anomalous instances. I think we should share the Nobel Prize for this one....

While our collective brain trust ponders the nature of triangles defined by three uniformly random points chosen inside a circle, specifically the mode of their areas, we ask another question. Recalling that the mean and median areal coverage of its circle's area are about 7.4% and 5.4%, what is the probability that a random triangle covers its circle's center?