when you say extended, i assume you mean uniformly stretched. if so, then yes. if it is just extended on the end, then no.

Nope. See my post above yours. The left side 1, the right side comes out to 100, so it evaluates as false.

The worm's location is not static, it moves as well when the stretch occurs. For example, at start the worm is 50 cm from center. After 1 second it is not 49 cm from center, it got pulled left due to the stretch... it's in fact just under 99 cm from center now.

1m long string, moving 1cm, stretching 1m

worm: 1 / 100
stretch: worm = .01 / 110 (something like that)
worm: 1.01 / 110
stretch: worm = .0101 / 120
worm: 1.0101 / 120
...

maybe under this exact example it is not gaining relative to the stretch. any of: a shorter string to begin, a greater movement, or a lesser extension and it would..

not sure, i'm not a math guy.

You're correct. I am a math guy, and I posted the equation to determine that a few posts up.

Oh, I've been reading this puzzle wrong this whole time.

Yeah, the key is that it states the worm is on the string.

This seems wrong by thought.

If the stretch is <= his movement then some piece of the stretch is always put behind him, which means that he always gains distance vs the stretch.
So he must be able to reach the end no matter the length.

You're getting the wrong answer here i believe.

You're right that I was a little wrong. I had originally started developing the equation as position from left, then changed it to distance from right, and that numerator got lost in translation. It's something like what you put but still not quite right. I'll figure it out.

d[t] = (d[t-1]-m) + s*((d[t-1]-m)/(d[0]+s*(t-1)))

The 1st parenthetical term is your new distance before the stretch.

The 2nd parenthetical term is the effect of the stretch. The numerator is your new pre-stretch distance, the denominator is the current string length. That ratio is your progress mark (between 0-1), scaled by the size of the stretch.

And the worm can reach the end if:

m > s*d[0]/(s+d[0])

and i'm spent

Mathematically speaking, the worm will reach the other end. It may take several times the age of the universe ( I haven't calculated for the exact description). The fraction of the length of the string travelled grows as a harmonic series, which is unbounded.

In the case of the 1m string, stretches 1m/s, worm moves 1cm/s, the advancement of the worm each second as a fraction is:

t1: 1/100
t2: 1/200
t3: 1/300
...

So the total progression is SUM[t=1 to INF] 1/(100t). I don't remember how to calculate that limit but surely it can't be >= 1?

edit: well I'll be damned. I googled Harmonic Series Diverges and under the section of Paradoxes the first example is this very worm/string problem, and the answer is yes, the worm will always reach the end.

https://en.wikipedia.org/wiki/Harmon...ics)#Paradoxes

sure it is

1/1+1/2+1/3+1/4+1/5...=infinity

so

1/100+1/200+1/300...=infinity as well

Yeah. I haven't touched most series-related stuff since Calc 2 like 20 years ago.

Yeah it is crazy 1/1+1/2+1/3+1/4=infinity

1/(1^1.01)+1/(2^1.01)+1/(3^1.01)+1/(4^1.01)+... does converge to a value

Is this basically saying that if I have a stick attached to my head and on the end of the stick there is an apple that is 1m in front of me. If I walk to the apple, it keeps staying 1m away, but if I walk for an infinite length (meaning the apple also goes an infinite length) that since infinite = infinite that I will eventually get to the apple?

No, it's not the same thing. Go back to the worm problem.

At time 0, worm position and string length are both 0. The string now stretches by 100 cm (to become the initial length). 100% of the stretch is ahead of the worm.

At time 1, the worm moves to position 1 cm. Then the string stretches 100 cm. But, only 99% (1 - 1/100) of that stretch is ahead of the worm. So the distance only grew 99% longer. The worm's position is now 1.01 cm.

At time 2, the worm moves to position 2.01 cm. Then the string stretches 100 cm. But only 98.995% (1 - 2.01/200) of that stretch is ahead of the worm. So the distance only grew 98.995% longer. The worm's position is now 2.0302005 cm.

At time 3, the worm moves to position 3.0302005 cm. Then the string stretches 100 cm. But only 98.98989933% (1 - 3.0302005/300) of that stretch is ahead of the worm. So the distance only grew 98.98989933% longer.

As you can see, the growth rate of the distance between the worm and the end decreases. Eventually (and I mean eveeeeeentually), the growth rate becomes negative, and the worm finally starts to gain ground... that gain will slowly increase until it finally reaches the end.

If you want to think of it in terms of physics, the worm's velocity is constant, but the string's forward stretch (velocity relative to the worm) is slowing down (it has negative acceleration), and eventually the worm moves faster than the forward stretch. The reason this is different from your apple/stick example, is that 100% of the apple movement is in front of you, so both you and the apple are moving at constant relative velocities.

^ ok, that's basically what I put in post 104. basically as the worm advances, less and less of the stretch is happening in front of it.

Yup. I didn't grasp that until after e_holle mentioned harmonic series. Wish I got that earlier in the day, would've saved me a lot of time churning out useless equations

Puzzle #7

You're walking from OP Mest to Miram Shah. It's the desert and there is no place to stop for water along the way. You have to take all the water you need with you. It's 10 days' walk and you can only carry 3 days worth of water with you. You can bring helpers with you and they can each bring 3 days worth of water with them. The helpers need to drink water too, but they don't need to come all the way to Miram Shah with you. If they leave you along the way then they must have with them at least enough water to get back to OP Mest. You can use as many helpers as you want and you can stash water along the way.

How many helpers do you need to get to Miram Shah?

Hint:

Answer for spoilered sub-puzzle to maybe help people get started

I suspect this isn't really the question that you meant to ask, but if it is then: