Time And Motion

gondorff

It's a mile back to my house from my nearest supplier of psychedelic wonders. I start walking and as I take my first step the distance doubles, and continues to double with each subsequent step.

Do I make it home?

2Scoops

No. You are arrested and brought to jail for taking mind-altering drugs.

gondorff

Arrested by whom? And why would they suspect me of taking mind altering drugs?
More importantly, would they catch up with me if, after every step they were to take, the distance between myself and themselves doubled?

Fremen

It's a mile back to my house from my nearest supplier of psychedelic wonders. I start walking and as I take my first step the distance doubles, and continues to double with each subsequent step.

Do I make it home?

Yes, but only if your first step is greater than or equal to two miles in size, or your stride length doubles every step.

(ask a weird question, get a weird answer...)

Delphi91

Fremen wrote: »

Yes, but only if your first step is greater than or equal to two miles in size, or your stride length doubles every step.

(ask a weird question, get a weird answer...)

No, the first step need only be 1 mile long but in the opposite direction to his house. That puts him two miles away.

You know, I hate these problems which look "bleedin'" obvious. I'm always afraid that there's a catch somewhere...

Pherekydes

Maybe he lives in a mirror.

gondorff

Lol. 'Not' a trick question.

Let's say my pace is, and will always be, one yard and I always make progress towards my house.

Incidentally, I don't know the answer to this. I remember reading something similar a number of years ago that went something like:

A fly starts crawling along a centimetre long rubber band, which stretches to double its length with every millimetre of progress the fly makes...

or something...

Fremen

Ok, if you want a straight-faced answer to this, then you just have to think about then number of steps left to get home after any given step.
Suppose when you're at the shop, there's 1,000 steps until you're home.
You take one step, so 999 left, but then the distance doubles, so there's 1998 steps still to go.
On your second step, there will be 3994 (1997 X 2) steps until you get to your house, and on your third, there will be 8986 (3993 X 2). You can see that the number of steps is increasing, so you can't get home.
If you wanted to be rigorous, you could prove this using induction.

Oh, and incedentally,

Let's say my pace is, and will always be, one yard and I always make progress towards my house.

Assume there are a finite number of other houses between your house and the shop. Now, some of these houses will have a yard. As the distance doubles, the number of yards stays constant, and the number of yards between you and your house decreases. Thus, if the length of your pace is one yard, you can eventually expect to get home.

gondorff

Fremen wrote: »

If you wanted to be rigorous, you could prove this using induction.

I would like to see this proof.

What you are suggesting, in essence, is that if I make steady progress towards a goal, I will never reach that goal.

In a similar vein, I think there exists a mathematical proof that parallel lines meet at an infinite point in projective space. I wonder if this proof, whatever its substance, could be disproved by induction?

Fremen

Let D(n) be the distance to your house once you have taken n steps. Then you can make it home only if D(n) = 0 for some number finite n.

Let D(0) = DIST be nonzero
Let STEP be your step size
Then D(1) = 2x(DIST - STEP)
D(2) =2x(D(1) - STEP) = 2X(2X(DIST - STEP) - STEP)

In general,
D(N) = (2^N)XDIST - (sum from i = 1 to N)(2^(i)XSTEP)
= (2^N)XDIST - (2^(N + 1) - 2)STEP
= (2^N)(DIST - 2XSTEP) + 2XSTEP

which is increasing if DIST > 2XSTEP

This says that you actually can make it home if your stride length is greater than half the distance to your house. Turns out you don't need induction at all.

The notion of parallel lines is very interesting. Euclid assumed they didn't meet in a claim known as "Euclid's fifth postulate". This has actually been shown to be independent of the other axioms of geometry. You can assume it's true or that it's false, but either way, you won't get a contradiction. In a sense, it's true and false at the same time.

seandoiler

Fremen wrote: »

The notion of parallel lines is very interesting. Euclid assumed they didn't meet in a claim known as "Euclid's fifth postulate". This has actually been shown to be independent of the other axioms of geometry. You can assume it's true or that it's false, but either way, you won't get a contradiction. In a sense, it's true and false at the same time.

leading us to non-euclidean geometries...