skypi,

1 second is a very long unit of time in computer terms... the 64 disk stack could be solved by fastest computer a lot quicker than 1 nanosecond per move.

Something tells me you are not really feeling the size of the problem in your gut.

Certainly we could kick out all the monks, shut down the monastery and get a computer to go though the motions of the Towers of Hanoi instead. A typical computer today might manage one move per nano second. That's pretty quick right?

Not really, it's a billion times faster than our monks were, it will get the job done in 200 years or so rather than 16 times the age of the universe as I calculated before.

But, all I have to do is make the problem a little bit harder, just double the height of the stack of disks from 64 to 128.

Boom, our new improved computer solution is not going to end until 16 times the age of the universe!

That is the depressing realization with these kind of problems. No matter how fast your machine is it soon gets bogged down if the problem is made just a little bit bigger.

That of course is why the crypto algorithms we use work. There just isn't the compute power available. When it looks like there might be as computers get faster what do we do? We double the size of the keys. Safe again.

As for the frog (or two lines of people) my summary would be, if a move must be taken each time, there comes a time where the minimum move possible by an object (frog or person or whatever) exceeds the theoretical desired distance to travel and the object must overshoot/collide with the target object with the next move.

Mathematicians might not worry about "the minimum move possible". Why would they? Whatever small thing you have you can always have a smaller one. That is how we think about real numbers.

Physicists on the other hand will worry about it. They will point out that quantum mechanics tells us that when things get really small we cannot know where a thing is. We won't know if the frog crossed the line or not, we won't be really sure where the line is.

Worse still, if you really try to measure down to really small distances, around about the Plank Length, 1.6 x 10-35 m, then the energy we have to put in is so huge that the whole experiment will collapse into a black hole.

Where is our frog now?

Memory in C++ is a leaky abstraction .