Once upon a time, I did the same.

Beat the os (to loading) that is interpreting the values, and you'll get your answer, for that environment only. You'll have to know the hardware capabilities, but more importantly its own code that is "burned in" to see how it manages things.

Edit to add: The Bios is a boogie man in between.

If I understand you correctly, this kind of thing is handled by the CPU microcode? I'm guessing I'll need to study modern assembly language and check the code of modern open source O.S. to really get it.

I forgot and amended my prior post just before you responded; Bios is in the way.

Good ones make, or break, the hardware from the user's perspective later on.

I played with assembly, way back, but quickly had to make a living using higher/canned languages later on.

Edit to add: Stack logic was the key in the early years, now it is just trying to get past, or use, the code that comes before us to do things that they don't.

1- why is Prince of Persia 1 (and others) free to play on browsers, but it's missing here? : (
2- Dwarf Fortress: isn't still possible to build multiple bases in the same world? Why creating a complex huge map, then?

how much wood would a woodchuck chuck, if a woodchuck could chuck wood?

A woodchuck would chuck as much wood as a woodchuck could chuck, if a woodchuck could chuck wood.

Why do the gog game and movie prices start at 6 USD but everything else on GOG seems to be in increments of 5 dollars?

My suspicion is that early in gog's life, someone read a chapter on psychological thresholds for pricing, and somehow, $5.99 and $9.99 are magic (that is, well-perceived) numbers.

Good thought 9.99 isn't 10 that is a well known marketing scheme. Yet 5.99 is just slightly akward or is it the 6 goes with the 60% standardish weekend promo?

I had to think for a while, but then realized that indeed for every odd n, the gamma function generates a sqrt pi that cancels out the most recent increase in the exponent of pi... quite astonishing.
Any idea where this gamma function in the formula comes from? The definition shouldn't appear so easily just by formulating the problem as an integral, since the g f requires infinite bounds... ?

The derivation on the wiki is inductive. That is, the volume of an (n+1)-sphere is some function of the volume of an n-sphere. So you can imagine that a 5-sphere volume might include multiplicative terms from the equations for 4-spheres, 3-spheres, etc. If these terms increase as n does, you'll wind up with something that looks like a factorial:
3-sphere -> 3x2x1
4-sphere -> n x 3-sphere -> 4x3x2x1
5-sphere -> n x 4-sphere -> 5x4x3x2x1
And the gamma function is pretty similar to the factorial function. That said, it's certainly not obvious to me how the gamma function generates pi terms. Magic!

(I've simplified the above to make factorial origins more obvious. You can see from the wiki that it's more complicated than I've made it out to be. Sorry if it's a clumsy answer, I'm unsure myself. And I certainly haven't tried integrating a particular dimension directly, instead!)

I found $5.99 awkward, too. I imagine it was determined experimentally and listed in a book somewhere.

The sqrt pi appears when you try to solve the integral that defines the gamma function. It's not so far away from the Gauss error function.

to understand why it is like it is, you will have to get through some university courses (which require a higher school education):

first you'll need basics in classic logic, simple set theory and number theory

then linear algebra (vector space theory, maps, linear maps and matrices, permutations, determinants maybe more)

then analysis 1, analysis 2, analysis 3 (real numbers, sequences, continuity, one dimensional differentiation, higher dimensional differentiation, basic topology, one dimensional integration, lebesgue integration, spherical coordinate system, measure theory maybe more)

to get through all of it you need 1 or 1,5 years, then you'll be able to see by yourself why the relation between dimension of sphere/ball and the potence of pi in the surface/volume formula is like it is. you'll be probably disappointed as there IS NO MAGIC, its more a matter of how you define coordinate systems and measures. additional its useful to know some theoretical physics like advanced mechanics, quantum theory and statistical physics.