Talk:The monkey and the coconuts

Solution by continued fractions
First, we don't show how to reduce the problem to the given Diophantine equation. That's not trivial, though the algebraic part pales compared to solving it once done.
 * Done.Sbalfour (talk) 17:43, 15 December 2019 (UTC)

The standard method to solve the equation is by continued fractions, which is not given in the article. It's bound to succeed, but a bit of work. That's what mathematicians do. It's fortuitous that one of the infinitely many periodic solutions is a small negative number, and some clever person found it by trial and error. That's not a deterministic solution, as by a slight modification to the original problem, I can obviate any negative (or positive) solutions smaller than $$|n|$$ where $$|n|$$ is infeasably large (in fact, that appears to be exactly what Williams did). Sbalfour (talk) 22:29, 14 December 2019 (UTC)
 * I've added the continued fraction solution. Actually, this is the only such place where I've seen a complete and replicable such solution.  I've tried not to skip too many steps so it's understandable by non-mathematicians.  Anyone who's had calculus or a course in number theory should be able to generate the continued fraction itself.  This is not the place for an exposition of the technique. Sbalfour (talk) 16:20, 16 December 2019 (UTC)

With due deference to Gardner, his statement (copied in a footnote in the text) that continued fractions are 'ingenious' but 'long and tedious', is disingenuous: continued fraction solution of Diophantine equations has been standard mathematical technique since the time of Lagrange; generating the 7-denominator continued fraction took me no more than ~20 minutes and three lines on a scratch pad. His admonition to go look up the technique in a math book probably discouraged generations of armchair mathletes from learning and applying a valuable technique. Hard to believe printing space was so precious that a footnote expanding the fraction would've been out of place.Sbalfour (talk) 16:20, 16 December 2019 (UTC)

Solutions and derivations
I've added quite a few solutions, and several of the ones that rely peculiarly on the recursive structure of the problem are TBD. I've tried to keep the solutions very brief, and avoid calculus, linear algebra, number theory, and other advanced topics and dense mathematical formalisms. This is an article that the public, who by and large know no mathematics at all, not even algebra, is going to read. And by and large, the mathematical acumen needed to solve the problem will therefore escape them. Just formulating a usable equation is challenging, because there seem to be more unknowns than equations for them (hah!). Applying the solutions is different than understanding why they work: the former should be understandable; the latter requires some training in number theory. Though Euclid's algorithm is purely arithmetic that could be taught in junior high, probably the public hasn't seen it, because probably, it isn't needed much until one gets to algebraic number theory.Sbalfour (talk) 22:12, 18 December 2019 (UTC)

In Gardner's book, he jumps from the "trick" negative and blue coconuts paradigms directly to a generalized formulation of the solution without ado. There's about a page of moderately complex calculus of finite differences between the problem and that solution. That's disingenuous to a non-sophisticated audience. Even a mathematician might stumble around for a while before discovering the derivation. It's not fun to work out that derivation. Though it's sourced to Gardner, and desk-checking it shows it's correct, I think, in line with the other solutions, we ought to show how to get from the problem to the solution.Sbalfour (talk) 16:08, 21 December 2019 (UTC)
 * Done. I discovered along the way that Gardner's formulation is different and rather unnecessarily cumbersome, so I've moved it to a footnote, in order to not clutter the presentation. The encyclopedia is not a textbook, and maybe we don't need Gardner's formulation at all.  With this addition, I think I';ve written finis to this article, and anything you'd want to know about The monkey and the coconuts.Sbalfour (talk) 18:30, 4 January 2020 (UTC)

Diophantine analysis sidebox
Originally, I decided not to include anything about Diophantine analysis in the text. However, the problem is fundamentally Diophantine in nature, and solution requires understanding what the solution space looks like. So various tidbits of Diophantine analysis got sprinkled through parts of the text. If an amateur puzzlist comes upon the problem with no knowledge of Diophantine analysis and appproaches the problem algebraically, they may give up before they even start, because they'll only be able to define one equation with two unknowns. The solution space of algebra problems is one equation per unknown and most people know that. So they'll think they're not understanding the problem in some way, because another equation must be there, or that the solution involves some kind of trick, or that trial and error is the only way to solve the problem. I first encountered this problem as a child, and that's what I thought. My child self would have been delighted by what my mathematician self would have written in the sidebox. So the sidebox describes the solution space for the problem.Sbalfour (talk) 20:28, 22 December 2019 (UTC)

C class and low priority
I've bumped the article from Start class to C class, and unassigned to Low priority. If anyone begs to differ, feel free to adjust. I'm considering now, after substantial additional text was added, bumping it yet again to B class. I think the article has 80% of everything that can be said about the problem, at least briefly. It's never going to be a full-length article- there's just not that much to say about it. The article is already a bit too much math, though I've kept it generally concise for the target audience of high school grad with a smattering of algebra.Sbalfour (talk) 22:12, 22 December 2019 (UTC)

original research and ref improve
These two tags are related. The content concerning in the Solutions section isn't really original research (I didn't invent any of the solutions - I wish I was that clever); several were taken from web sources of dubious authority like blogs or personal websites. The solutions of a math problem can be verified, by solving the problem accordingly and plugging in the resulting numbers, so they're good. But citing Joe Blow's blog won't cut it per WP:RS. I don't have access to journals via Jstor, ResearchGate, Science Direct, etc. so I can't provide inline citations at this time. Someone who does, can fix many of these up.Sbalfour (talk) 19:35, 24 December 2019 (UTC)

History
The history is much richer than herein annotated: there were Babylonian as well as classical Egyptian sources for problems in the same class. The monkey + coconuts formulation appears earliest in Carroll's work; earlier formulations were divisions of flowers, fruit, animals, and other objects. The modern ending where a final division comes out even appears in centuries earlier renditions. So the problem passed through many hands while evolving to the William's version. Number theory was in its infancy in the 18th thru the 19th centuries, and this kind of problem was fodder for its development. Many of the sources appear in David Singmaster's 8th edition of his "Sources in Recreational Mathematics", 2004. A comprehensive history leaves the realm of popular recreational mathematics and enters into research in the history of mathematics, so what's reasonable for this section is limited to highlights in that area.Sbalfour (talk) 18:26, 27 December 2019 (UTC)

OR flag
I removed the "Original research" warning. Articles on mathematics are bound to be edited by mathematicians, who love thinking above all. WP abhors thinking, in favour of "copying from sources", but thinking is bound to pervade such an article. So there has to be a very specific claim that some statement in the article is the result of thinking, but is not from a "source" somewhere. Then we can argue, and attempt to explain to antithinkers why it is obvious. Imaginatorium (talk) 05:07, 27 September 2023 (UTC)