@Carl Love Please see my response to Kitonum. My knowledge of the Gauss-Bonnet theorem is too thin to answer your question. 

@Joe Riel I tried to create a smaller example but DEBUG in it did not display its values in the worksheet, so below is the full worksheet in which I found the problem. The DEBUG values at the worksheet's bottom were left there after using continue to display 3 debug windows.

I use DEBUG at the top level to show values during several loop iterations and to halt loop execution.

DEBUG_test.mw

The uploaded worksheet below (in Maple 2016) contains an ODE with three events. These examples of event coding may help your understanding.

Reverse_spin_cueball.mw

@Kitonum It seems the volume of this body also equals 1/24

plot3d(x*y, x = 0 .. -1/3, y = 0 .. -x+1, style = surface, filled);

since int(int(x*y, y = 0 .. -x+1), x = 0 .. -1/3) = 1/24

@Rouben Rostamian  Once again I am in your debt. Is it true for any two 3D surfaces which are tangent to each other that their normals at their common point of tangency are collinear?

@Rouben Rostamian  Thank you for taking the time to answer me. Since my math skills are about middle undergraduate this subject is interesting but far over my head.

@Carl Love When I execute the 4th method I receive "Error (in Last) too many levels of recursion".

The following code works in Maple 2016

Last := proc (k::posint)

procname(k) := 4*thisproc(k-1)+k :

end proc:

Last(0) := 1:

`seq`(Last(k), k = 1 .. 10);

@Christopher2222 are enlightening. It is fascinating that this question has drawn the interest of a number of early calculus stars.

@Rouben Rostamian  your reference and the animation. Christopher 2222's references are also interesting.

@Carl Love I am always happy to expand my understanding of Maple's myriad capabilities and appreciate your helping me to do so.

@vv You show an interesting alternative. Are there cases where command inequal is generally superior to (or inferior to) the implicitplotting of an inequality? 

@Kitonum You show that plotting the function lhs-rhs created from the inequality lhs>=rhs illustrates the logic of implicitplotting of the inequality. Thanks to you I now have a better understanding.

@Rouben Rostamian  Thank you as always.

Below is a worksheet which finds the values of all of its ODE's variables at the time of event firing.

Eventfired.mw

@Rouben Rostamian  I am delighted that you continue to be interested in my question and I appreciate your advice re improved programming.

Before tackling problem 61 I would have expected Slinky's loops to be further apart as centripetal force increases with distance from its fixed point, but the math doesn't lie.

It seems you already had the "200 More..." book in your library. It and its predecessor are an excellent source of physics challenges. I look forward to your help when I am stuck on future problems. 

@Rouben Rostamian  This is great! Your worksheet above has enabled me to display Slinky extended by the centrifugal force of rotation.

Please check the code I have added to yours in the worksheet below. I will appreciate any comments you care to make.

Slinky2.mw 

If the above is accurate, I will proceed to animate the rotation.