The worm problem was introduced in the same message in which a reference
was made to Piaget. I must still be at the "concrete operational level"
because I was not able to follow Uri's explanation (my qeustion and his
reply are shown below). Note that the problem was presented in the
iterative way. Can somebody explain again how the cumulative solution
algorithm can be derived from the original wording (also shown below).
I think something is missing between the original wording and the
example Uri gives. Where is the reasonning which translates words into
As you can see, I was stuck before the "then open it up ... etc.". We
are addressing pedagogy now. I did use pencil and paper but this did
not help. I know that some of you smiling now; perhaps remembering a
joke. I do not mind this.
......................................................................
AM I THE ONLY ONE WHO DOES NOT KNOW HOW TO TURN WORDS OF THIS PROBLEM
INTO A CUMULATIVE EXPRESSION? Please answer this question by writnig
to me personally. Choose one of the following options, for example,
write "Count me in the category B", I am from Australia".
Category A - I did solve this problem without any help
Category B - I did solve this problem after being helped
Category C - I am still trying
Category D - I tried but gave up
Category E - I saw the problem but was too busy to work on it
Category F - I missed this problem
Category G - I am a phys-Ler, but not in one of the above categories
I will summarize the result of head counting in about a week or two.
ONCE AGAIN, THE CATEGORY ANSWER TO BE SENT TO ME, NOT TO PHYS-L
*************
(Why? Just imagin what happens when everybody sends a message to
everybody else on a large list.)
Ludwik Kowalski kowalskiL@alpha.montclair.edu
........................................................................
********************** QUESTION TO URI ***********************
Hi Uri:
I finally looked into your posted solution (of the Leigh's worm problem).
I compared your non-interative expression for the distance covered by
the bug in n steps
x{n} = (n+1)*(sum from k=1 to k=n of (1/k))
with the iterative formula I was able to derive (posted yesterday).
x{n}=[x{n-1}+1]*(1+1/n)
They lead to identical answers. How did you reason to obtain your
formula from the wording of the problem? How did you recognize the
harmonic series (1+1/2+1/3+1/4+ ...+1/n) in it?
******************** URI'S REPLY *********************
Ludwik,
These things are easier seen on paper (I just looked and recognized it),
than on e-mail where one is limited with fonts and symbols. Let me try,
though: write it out in detail, for a few cases (n=1, 2, 3, 4 ,etc.) and
you easily deduce the pattern for a general n, to be:
Write it on paper - so much more convenient. Then open it up, going from
right to left. Do you see what happens? you have (n+1) multiplying
everything. Then you have 1/n multiplying two terms: 1 and everything else.
1/n times "everything else" is 1/n times n/(n-1) which gives you 1/(n-1),
and that now multiplies 1 plus "everything else" again. You see how awkward
I am becoming in my explanation, but if you just write it down you will see
it immediately. You continue to open the parentheses, and the result pops
out as I wrote it down. I hope this helps.
***************** ORIGINAL PROBLEM FORMULATIN ********************
You are provided with an elastic rope exactly one kilometer in length. At
time t=0 a worm starts to crawl from one end of the rope to the other at a
speed of one centimeter per second. The rope itself is attached to a tractor
which is stretching the rope at a rate of one kilometer per second. At the
end of one second the worm has marched one centimeter toward the other end
of the rope. At that instant he is instantaneously transported to the 2 cm
mark on a rope which is now two kilometers in length. By calculation you
will see that at the end of the second second he will be three centimeters
along his way on the two kilometer rope, which will then lengthen to three
kilometers, taking the worm to the 4.5 cm point, etc. How long will it take
the worm to reach the end of his rope?
*************************************************************************