Tag Archives: Snail Problem

Snail Problem, continued

More about the Snail Problem posts by Darmok. Some math, and some background about the problem. This post is based upon Darmok’s most recent posts Snail Problem, Parts 4 and 5. Because I want to use previews, to check the Latex code before publishing, I’m not using the WordPress reblog mechanism. I’ll give links to all five of Darmok’s posts, and to my two prior posts on this topic, and to the several resources found, later on towards the end of this post.

MATH:

First some math. Given a twig of initial length $L_0$ growing at a rate $g$ , so that its length at time $t$ is $gt + L_0$ . A snail is crawling along the twig with position $x(t)$ at time $t$ . The snail starts at the base of the twig at time 0, so that $x(0) = 0$ . The problem (restated) is to find an expression for the position $x(t)$ of the snail, and to determine the time $t_f$ when the snail reaches the end of the twig. Darmok has solved this in his posts (links below). My earlier note on this had some errors.

The solution is
$\displaystyle x(t) = {{s}\over{g}} \, \big( gt+L_0 \big) \,\, \mathrm{log} \Big( {{gt+L_0}\over{L_0}} \Big)$

The time at which the snail reaches the end of the twig is finite, and given by
$t_f = L_0 \, \Big( {{e^{g/s}-1}\over{g}} \Big)$
or
$t_f = \Big( {{L_0} \over {s}} \Big) \, \Big( {{e^{g/s}-1}\over{g/s}} \Big)$
I like the latter form because it emphasizes the appearance of an $(e^u -1)/u$ structure in the solution. The factor ${L_0}/s$ is the time the snail would take to traverse the twig’s original length, ie if the twig were not growing. It is multiplied by a factor $(e^u -1)/u$ , where $u = g/s$ is the ratio of the twig’s growth rate to the snail’s proper speed, representing the expansion of the snail’s travel time.

Inverting the equation for $x = x(t)$ for the snail’s distance travelled in terms of time, produces an equation $t = t(x)$ for the time the snail takes to travel a distance $x$ . That expression involves the Lambert W function, as previously mentioned. I am going to reserve the details for now. My previous post on this had an error — it is Lambert W, but not the expression which I posted. I want to check my new formula.

However, it may be of interest to remark that the Lambert W function is closely related to a geometric curve, the quadratix of Hippias. The quadratix can be used to solve various problems, in the style of the Greek geometers (one of whom was Hippias, of course). It offers the possibility that the quadratix of Hippias could be used as part of a geometric solution of the snail problem. That might make a very nice diagram.

BACKGROUND:

Now for some background about the snail problem. The only mention of this problem in its “snail on a twig” variant that I have found on the net, is a quiz in Alexandre Kirillov’s advanced calculus course in fall 2012 (links below to the quiz, to his teaching assistant Zhaoting Wei’s website which has the quiz posted, and to A. Kirillov’s home page). I wrote Kirillov, and he told me that the problem is “part of the mathematical folklore of the Math Department of Moscow University” in the 1950’s and 1960’s, and could be found in some lectures, handouts and books written by Moscow mathematicians. I have not yet gone looking. You also might enjoy that search. One possible resource which Kirillov suggested is the Arnold problem collection “Trivia” published in Uspekhi Mat. Nauk. Upon further inquiry, Kirillov also wrote me that he “certainly knew this problem before 1972 and it could be much older”.

There is a Wikipedia article (link below) of the same problem, described as an ant marching along a rubber rope. That article mentions the problem appearing in one of Martin Gardner’s “Mathematical Games” columns which appeared in Scientific American magizine. I don’t know which year/issue. However, that column has been reprinted in Gardner’s book “Time Travel and Other Mathematical Bewilderments”, Freeman, 1988, pages 111-112 and 118-119, and that book is sitting in front of me at the moment. (Thank you to local public library!) The chapter title is “The Rubber Rope and Other Problems”, which is probably also the subtitle of that instance of Gardner’s Mathematical Games column, if you want to search the Scientific American article database.

Gardner says “This delightful problem, which has the flavor of a Zeno paradox, was devised by Denys Wilquin of New Caledonia. It first appeared in December 1972 in Pierre Berloquin’s lively puzzle column in the French monthly “Science et Vie”.

I will leave it up to others to pursue questions of priority, as priority is not my concern. I am more curious about the intellectual history behind an interesting problem. My own stance is to rely upon Kirillov’s recollection, that the problem was extant in the 1950’s or 1960’s.

Good problems, like folk tales, float around in our collective conversation and acquire various guises as noun substitutions are made (snail, ant, caterpillar etc) and, more significantly, as new contexts are imagined. The choices of nouns and contexts are made to highlight some aspect which is of particular interest to the audience. The dynamics, the verb phrases of the problem, the processes which are involved with its exploration — those are more important than the nouns and the contexts.

I imagine that the snail problem might have even older appearances, for instance perhaps in the papers or correspondence of the mathematicians of the 1800’s, as it is the sort of enjoyable and intriguing puzzle that would have entertained many over the ages. The snail problem has relationships to Zeno’s paraoxes, and to the limiting arguments of Archimedes, and as noted above, possibly has a geometric solution related to the quadratix of Hippias.

Perhaps I should mention that the solution in Gardner’s column was not the differential equation approach found by Darmok, though of course the solutions are comparable. The one which Gardner gives is a series argument, based on the relative distance $x(t)/L(t)$ travelled by the snail (ant). It relies upon the divergence of a uniformly distributed subset of the terms of the series $1 + 1/2 + 1/3 + ... + 1/n + ...$ to establish that the snail can reach any relative distance, hence can get to the end of the twig (rope).

That’s all for today! You will find plenty to explore in the links below.

Best wishes,
Ken Roberts
01-March-2014

Useful links:

The five snail problem posts by Darmok:

Snail Problem, Part 5

Snail Problem, Part 4

Snail Problem, Part 3

Snail Problem, Part 2

Snail Problem

My two prior posts on this topic:
https://lasi2.wordpress.com/2014/02/24/snail-problem-part-3/ (and see comments)
https://lasi2.wordpress.com/2014/02/23/snail-problem/ (and see comments)

Info about the quadratix of Hippias:
http://mathworld.wolfram.com/QuadratrixofHippias.html

Alexandre Kirillov quiz which mentions snail problem, and related pages:
http://www.math.upenn.edu/~zhaotwei/math_360_fall_2012/quiz_0.pdf (the quiz itself)
http://www.math.upenn.edu/~zhaotwei/math_360_fall_2012.html (course webpage by Zhaoting Wei)
http://www.math.upenn.edu/~kirillov/ (Alexandre Kirillov home page)

Wikipedia article about ant on a rubber rope:
http://en.wikipedia.org/wiki/Ant_on_a_rubber_rope

Discussion of cosmological aspect of problem:
http://cosmoquest.org/forum/showthread.php?72930-The-Ant-and-Rope-thing

Snail Problem, Part 3

4 Replies

For Darmok … comments on your Snail Problem part 3 post. Very nice! I confirm your results, both by checking the differential equation solution, and by calculation. Suppose that D is the original length of the twig (using D instead of L-zero as in your post), that g is the growth rate of the twig, and s is the crawl rate of the snail (relative to the twig). The snail will always get to the end of the twig, arriving there at time (1/g) * (exp(g/s) – D). The twig length at that time will be exp(g/s). That is, the twig length at snail arrival will be independent of the starting length of the twig!

The formula for x(t) = the distance crawled by the snail by time t, is (s/g)*D*((gt+D)/D)*log((gt+D)/D). I have written the x(t) formula in that form, (s/g)*D*(w*exp(w)) where w = (gt+D)/D, in order to emphasize the w exp(w) structure. Take a look at the Lambert W function — paper by Corless, et al, 1996 — or I think there is Wikipedia, and other info online. Lambert W shows up a variety of situations. It is a cousin of the logarithm (which I write log, by the way, not “ln” because that is hard to make out in typewritten info). w = log(z) is the multi-branch function which satisfies exp(w) = z. Similarly, w = W(z), called the Lambert W function, is the multi-branch function which satisfies the equation w*exp(w) = z. There are nice mappings between the z-plane and the w-plane, and they can be used to make seemingly intractable problems have very simple solution sets, if visualized in the other plane.

Hope this info helps. Have greatly enjoyed this problem.

Now, I am going to get wild… Twigs do not grow uniformly, they grow at their end. The stuff that grows uniformly is expanding balloons, or rubber bands, or the universe. I’ve long wanted to find a reasonable way to interpret the distribution of gravitational information (structure) at a faster speed than light. This snail and twig problem is illuminating, in that a slower information will still eventually arrive. Just arm-waving, no serious math, but something that I’ll store away for a revisit someday…

Best wishes,
Ken Roberts
24-Feb-2014

Ancora Imparo

This is a continuation of my attempt to solve the following problem: A snail crawls along a twig of length L at 1 cm/d, and the twig grows (along its entire length) at 2 cm/d. Will the snail reach the end, and when? You can see part 1 where I successfully set up the differential equation and unsuccessfully tried to solve it, and part 2 where I did manage to solve it.

At the end of the last post, I had solved for the equation that describes the position of the snail over time,

$latex displaystyle x = left(t+frac{L_0}{2}right)lnleft(frac{2t}{L_0}+1right) $.

Also, recall that the length of the twig is

$latex displaystyle L = 2t + L_0 $

since it grows at 2 cm/d. We want to find the time t when these two are equal.

$latex displaystyle 2t + L_0 = left(t+frac{L_0}{2}right)lnleft(frac{2t}{L_0}+1right) $

[Edit: I missed a very obvious step…

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Snail Problem

6 Replies

This is a nice pair of posts, by blogger Darmok at Ancora Imparo, about a snail crawling along a twig. Suspend disbelief for a motion, and suppose the twig grows uniformly (along its entire length), at 2 cm/day, whereas the snail crawls at 1 cm/day. Will the snail ever reach the end of the twig? Darmok has been working on a differential equation, of distance x(t) which the snail has covered at time t, compared to length L(t) = L0 + 2t of the twig. I’ve played with a variant, looking at the distance y(t) to end of twig. There is probably a nice graphical way to see the answer without using differential equation. The answer requested is a simple Yes/No, about whether the snail reached the end of the twig, not the time of arrival. Anyway, you might enjoy this problem too.

Best wishes, and thanks to “Darmok”,
Ken Roberts
23-Feb-2014

Ancora Imparo

A friend mentioned this problem a while back, and while I was able to use a computer to get an accurate numerical solution, I really want to find an analytical solution using math. The problem is that it will require differential equations, which I no longer remember. Edit: I haven’t yet been able to solve this problem, but I’m preserving my initial attempts below. See Part 2 where I made more progress.

The problem is this: a snail starts crawling along a twig, from one end towards the other. The snail can crawl at 1 cm/d. But the twig is growing at 2 cm/d. The twist is that the twig is adding material along its entire length, so the snail will be carried forward a bit, too, depending on where it is. Will the snail ever reach the end? Let’s let L be the initial length of the twig.

Let’s…

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