@ecterrab That's impressive work on the part of AI.  It explains the step-by-step calculations quite well and every step checks.  What is missing, however, is the motivation behind those calculations.  That's where the human touch is superior in comparison to the current state of AI.

A very nice analysis of the problem is available in https://arxiv.org/pdf/2505.19088 and provides some interesting ideas behind the solution.  It gives 3 different parameterized solutions; see sections 2.2, 2.3, and 2.4.  It is not clear whether those 3 provide all possible solutions.  It will be interesting to see if the AI's solution generates numbers beyond those given by that paper's formulas.  I haven't checked.

Hello Edgardo,

I can't tell whether ExactAI is limited to Physics, or any CAS-related questions.  If the latter, then you may be interested in the following.

I passed this to ChatGPT:

Are there positive integer solutions (a,b,c,u,v,w)
to the following system of equations?
                       a + b + c = u^2 
                        ab + bc + ca = v^2 
                        abc = w^2

It returned one valid solution.  (That's an improvement over last week when it said
there are no solutions.)

Then I asked the same question with an extra sentence at the end:

Are there positive integer solutions (a,b,c,u,v,w)
to the following system of equations?
                       a + b + c = u^2 
                        ab + bc + ca = v^2 
                        abc = w^2
If yes, are there infinitely many solutions?

ChatGpt tried various techniques, and finally concluded that 

"Therefore the system has no positive integer solutions.".  See the attached transcript in response.pdf which contradicts its previous finding.

I don't expect an AI to have answers to all questions, but it is disconcerting when it states a wrong answer as fact.  Perhaps your ExactAI can prevent such incidences.

As hard as it is to believe, the solution x=1633780814400, y=252782198228, z=3474741058973 was actually obtained by Euler himself without help from Maple or even a calculator ;-)

But that's not the smallest solution.  The triplet x=45, y=64, z=180 is also a solution.  There are a lot of other solutions in between.  

By the way, both ChatGPT and Google's Gemni claim that no such x, y, z can exist!

Spoiler alert!

Have a look at https://arxiv.org/pdf/2505.19088 to see how it is done.

I use Maple quite a bit in my daily work, but like you, I find the ribbon interface to be slow and inefficient, and that's why I am staying with Maple 2024.  I would really like to see that former interface returned as an option.  Your suggeston of a user-customizable interface will work for me too.

@janhardo That's the classic isoperimetric problem.  Here is how we solve it in Maple.

Download isoperimetric.mw

@janhardo Yes, it is trivial if you recall the definition of arcsin.  That function takes an argument between -1 and 1 and returns an angle between -pi/2 and pi/2.

Thus, we define

theta = arcsin(y'/sqrt(1+y'^2))

or equivalently,

sin(theta) = y'/sqrt(1+y'^2);

where theta is between -pi/2 and pi/2.

@Alfred_F I have cleaned up my solution to the extent possible and will post it now.  There are some elements that overlap with dharr's solution, but there may be some instructive aspects in it as well.

@dharr I am glad to see that our answers have converged.  I am going to post my solution once I have cleaned it up just to provide an alternative approach.

@Alfred_F My calculation shows a teardrop-shaped domain whose value corresponding to the fence length of 100 is 15468.56. 

I will post the solution if I find some time to clean it up.

@dharr If I am reading your work correctly, you are concluding that the optimal shape is something like a teardrop with a flattened edge on the right.

That flat edge looks suspicious.  In the figure below, we remove part of the domain from the top and add it to the flat side. That does not change the fence's total length.  The removed land lies between zero and xmax, so it is only moderately priced, while the added land lies to the right of xmax, so it is highly priced.  The change, therefore increases the value of the integral.  So the original integral could not have been maximal.

@erik10 I don't understand what you mean by an "outline of a sphere" or its "outer reach".  See if you can clarify that.

Do you perhaps want to project the sphere onto a plane?  If so, then plottools:-transform is the right tool.  For instance, to project the sphere (or any 3d drawing) onto the xy plane, name the drawing, say call it p, and then do

plottools:-transform((x,y,z)->[x,0,z])(p);

@erik10 To get a smooth sphere, you may draw the sphere in the usual resolution without a grid, and then superimpose the desired grid, as in:

plot3d(1, t = -Pi .. Pi, p = 0 .. Pi, coords = spherical, style=surface);
plot3d(1.01, t = -Pi .. Pi, p = 0 .. Pi, coords = spherical, grid = [25, 19], style=line, color=black);
plots:-display(%%,%);

Note that the grid is drawn at a slightly larger radius so that it does not get blended with the surface.

PS:  The grid drawn by Maple consists of straight line segments.  If you want properly curved grid, then it can be done through plots:-spacecurve.

@aroche Thank you for prompt action on this matter!

@acer Thanks for the extra work.  Regarding the automatic coloring of different curves by display(), yes, I did see the redraw=false option noted in the help page.  I am not quite sure of its purpose, since setting the color option overrides the default colors.

Here is yet another curiousity.

My original example noted that the command

plots:-display(pic1, pic2, scaling=constrained);

plotted the diagram over an incorrect range.  Interestingly, specifying a color option fixes the issue and the curves are plotted over the correct range!

plots:-display(pic1, pic2, scaling=constrained, color=[red,green]);

So that's yet another workaround.

@acer Thanks for your comments and your offer of submitting a bug report.  Please do.