[VA] SRC #011 - April 1st, 2022 Bizarro Special

[Valentin Albillo]

  
Hi, all,
                
Well, my HP calc site reached the 100,000 downloads mark a few days ago and on top of that today it's April, 1st, so let's celebrate the occasion ... (drum roll) ...

Welcome to my SRC #11 - April 1st, 2022 Bizarro Special

intended to once again put your brains and your HP calculators to work, this time featuring a bizarre challenge which nevertheless has important, useful real-life applications. But first, some exposition (Note: What follows are mostly my own ramblings © me, DC Comics are not to blame !):

The seasoned veterans among you might fondly remember the classic Superman comics of the '60s, and that within the Superman universe there was the Bizarro World (also known as Htrae), a fictional planet home to the eponymous people, whose society is ruled by the Bizarro Code, which states:

"Us do opposite of all Earthly things !  Us hate beauty!  Us love ugliness !  Is big crime to make anything perfect on Bizarro World !"

Originally a normal planet, the Bizarro World is now cube-shaped because the Bizarros couldn't stand the perfection of its spherical shape and Bizarroformed it like this:

In time, the Bizarro society thrived to the point where their cubic-shaped planet was quickly becoming overpopulated, so they set upon themselves the huge task of attaching face to face a second almost-identical planet to their original one, thus nearly duplicating the habitable surface and best of all, augmenting the imperfection of their homeworld's shape relative to perfect sphericity by making it prismatic, like this:

However, they first needed to make sure that the project was feasible, in particular that the gravitational force F between the planets when they were in contact would be manageable. Therefore, they commissioned Lebon Prize laureate scientist Rd. Nitnelav Albizarro #1 to carry out the computation, who immediately set to the task of finding out F when their respective centers were initially separated by an arbitrary distance d, which would then be shortened until the planets were in contact face to face.

To simplify matters and as the results could be easily rescaled afterwards, it was assumed that the gravitational constant G was 1 (in some units) and the planets were homogeneous cubes of side 1 (ditto) and mass 1 (ditto), initially placed like this:

Now, Rd. Albizarro considered a pair of sample points, (x₁, y₁, z₁) in Htrae 1 and (x₂, y₂, z₂) in Htrae 2, knowing that their contribution to the overall force would be:

and duly taking into account the symmetry (which nullifies the force exerted in the y and z directions) and integrating over all possible values for the respective point coordinates, quickly got this sextuple integral for the value of the force F between the planets:

to be numerically evaluated for the particular case when both planets are in contact, i.e. the distance d between their centers is 1.

As it happened, Rd. Albizarro was a Code-abiding citizen of the Bizarro society and wouldn't go for an unnecessarily accurate (ugh!) result but would instead be satisfied with obtaining about three correct digits. To that effect, and aware of the need to use some computing device in order to meet the assigned deadline, Rd. Albizarro promptly proceeded to check out the ones available in ugly human Earth at the time (ca. 1982) and saw that computers were so perfectly fit for the job that using one would be disgusting and could result in being severely reprimanded or worse, so the alternative was to get instead a programmable calculator (progcalc for short) as non-programmable ones certainly wouldn't do.

Restricting thus the search to progcalcs, it was soon apparent that there were essentially two main contenders, the ones branded TI (probably standing for "Totally Ideal", ugh !), and the ones branded HP (possibly standing for "Hardly Perfect"), the latter being just what was needed !

Now there was the question of selecting which particular HP model to use but that was easy-peasy, just a matter of looking at the specs and chosing the least capable one, which happened to be the HP-10C, the proverbial runt of the litter, a severely limited model having only 79 bytes of RAM available for storing programs and data, no subroutines, no flags, no indirection, no loop intructions, just two conditional tests, a meager function set, no I/O, and very slow to boot ... indeed, the least perfect model for the task at hand.

But Rd. Albizarro was unfazed, thinking that "If ugly Earth's Newton could do his gravitational calculations using this thing, it'll do for me as well", and against all odds actually succeeded by quickly writing a clumsy RPN (Really Perfect Not) program for the HP-10C, keying in the pertinent inputs, pressing the [R/S] (Run Slowly) key, going out to have a quick dinner and lo and behold, upon returning the computed value of F was already waiting in the display, which indeed was correct to three decimal places, as required:

---

Well, once the story's been told, it's your move:

Try to emulate Rd. Albizarro's achievement and write a program for the HP-10C which computes in a reasonable time a numerical value for the above sextuple integral correct to at least ~ three decimal places.

That failing (shame on you !), see if you can do it using other more perfect (ugh!) models, preferably HP and preferably vintage.

Note:  No cheating whatsoever allowed. Also, doing symbolic manipulations (transformations, dimensionality reduction, changes of variables), either by hand or using any CAS and/or giving math lectures is a big crime, you'll get arrested or worse, so stick to purely numerical computations. This is my last SRC for a looong while so don't spoil it for me, Ok ? Thanks.

I'll post my Original Solution for the HP-10C with results and extensive comments within a few days ... Or not, after all this might be an elaborate April Fools' Day practical joke !!   

V.