@Jean-Michel 

...Mr. Peano is to blame ;-) . His final formulation of the axioms named after him defines zero as a natural number. And for high school students and undergraduates, this is entirely sufficient and practically applicable. Naturally, in so-called "higher mathematics" a class-based definition grounded in set theory is employed. However in the otherwise excellent book *Exposé moderne des mathématiques élémentaires* by Lucienne Félix (page 553) zero is omitted during the introduction of the natural numbers, and the sequence begins with 1. It is not until the introduction of the integers that zero necessarily makes its appearance. In this respect, the book reverts to the original form of Peano's axioms.

@vv 

It works. Which type of "input" did you use? I would like to learn more about the different input methods.

@vv 

...I have made an inquiry. Unfortunately, after copying your algorithm, I encountered an error message, as shown in the attached file "test1." I am unable to locate the source of the error and would therefore like to request permission to copy a working version. Naturally, I fully respect copyright laws.
Your program performs a numerical sorting operation. It is a fascinating challenge that simply begs to be tested out.
I came across this problem in a delightful biography of Euler (Rüdiger Thiele, "Leonhard Euler", Teubner-Verlag, 1982, p. 158). The book offers a detailed description of Euler's teaching methods, his modes of thought, and his working style. Regarding this specific problem, the biography notes that Euler nearly despaired while attempting to solve it.
And since my inquiry has once again branched off into a broader discussion, I will take the liberty of adding the keyword "numeri idonei" here as well. ;-)

test1.mw

@vv 

...I wasn't familiar with that yet. I only knew that limitations on chord- length exist. But on page 408, this is now clarified. Many thanks! :-)

@dharr 

...the true intent of the puzzle has been sidetracked into a discussion regarding the rigor of Holditch's original proof. Naturally, with the aid of methods from convex analysis, it is now possible to provide a "rock-solid" proof—even for a generalization—of this theorem. The actual aim of the puzzle was, rather, to serve as a reminder of this elegant theorem and—perhaps with the help of Maple—not only to perform a calculation of the area (and thus of Pi) but also to generate an animation. It would be interesting to conduct numerical-graphical experiments starting from various initial curves, all of which invariably yield Pi. This approach could be utilized in the classroom to introduce differential and integral calculus, or—in more advanced semesters—within the context of functional analysis/Theory of Functions. Regrettably, however, I have long since ceased to be involved in teaching.

@vv 

To solve the puzzle, it is not necessary to invoke the theory of convex bodies or support functions. Since You have already "given away" the result, allow me to point out a simpler theoretical background—the almost forgotten theorem of Holditch.

@dharr 

...the solution path of the ODE simplifies if the initial equation is differentiated once. Assuming y'(x) is not equal to zero, the result is "test".

Download test.mw

@sand15 

"The largest octahedron (up to some symmetries) is necessarily the octahedron whose the base is the largest square S  contained in the cube."

... but not sufficient, it remains to be shown that the two octahedron vertices/tips land where they have to belong. However, this can be easily verified by calculation. Prince Rupert's cube is an interesting connection. That was news to me. It is also interesting that the cube in which the maximal octahedron is situated in a dual position has an edge length of 3/2.

@dharr 

And in the end, we get "only" the simple number "9/16". Interestingly, the dual cube corresponding to this maximum octahedron has an edge length of 1/2. Their solution would probably be nearly impossible with pen and paper. It demonstrates once again how powerful software leads to a clear, logical solution. My solution at the time relied on logically grounded limitations of possibilities and, in some cases, rather daring assumptions arising from symmetry considerations.

By the way:
I was able to learn a great deal about "geom3d" from your solution. Thank you for that.

@dharr 

There is another orientation for the octahedron that yields a slightly larger volume. Based on your result, the horizontal axis of the octahedron—whose endpoints lie at the midpoints of the cube's edges—should be tilted slightly. A volume of 9/16 is attainable, assuming I haven't miscalculated during my trial-and-error attempts.

@sand15 

...by creating an "artificial" sample. This works quite well for dividing into 1/100. I haven't worked with the "seq" command before. Learned something new again.

@acer 

Yes, I'm looking for the "distance" between two functions over their common domain, i.e., a global minimum, because I want to study the "practical" convergence behavior of sequences of functions, taking into account metric-related convergence principles. I've downloaded DirectSearch. The table of contents is very interesting :-) . Files were only found for the lib directory in the older version. The 2017 version doesn't offer any files for installation/inserting into "lib". In both cases, the help text isn't accessible.

Your reference to "inert Int" is important to me. With it, I can, for example, use the L_2 metric. There are just so many Maple intricacies I still need to learn – the difference between theory and practice.

@dharr 

I overlooked the mistake with the 50 (embarrassing). For practice with isoperimetric problems, I'll probably have to find simpler examples. In any case, the one in "test" is still a bit too advanced for me. Strangely enough, the approximate solution in Mathcad 14 was quite easy back then. Only finding a suitable function approach was somewhat time-consuming. I've included the old approximations in "test2".test2.mw

Download test2.mw

@sand15 

...I had hoped to define the search domain using initial points. Even after consulting the help texts, I still can't grasp the definition of the search scope and especially its syntax. I would appreciate any help.

Download test1.mw

@dharr 

"minimize" ?