The Mad Matrix

A Game of Choice

This still isn't exactly what I was looking for, but for the time being...:

Caveat:

I have no idea if this is valid enough to serve as a pattern or rule, or if it's just a function of pick- and- choose selection or simple coincidence.

In other words, I have no idea if there is a counterexample that would show the opposite, or otherwise inconsistent results.

The problem or task:

If given three known factors A, B, and C, is there a "blind" mechanical method for testing which of several possibilities of relation would be valid, and which would not?
And how many possibilities of relation are there at all?

The way I see it, the factors A, B, and C can be connected by multiplication in three distinct ways:

A=BC, B=AC, or C=AB

i. e. one as the result of the other two multiplied.

Expressing that relation by division is just a transformation or resolution of the above, in order:

A=BC, A=B/C, A=C/B

B=AC, B=C/A, B=A/C

C=AB, C=A/B, C=B/A

So all in all there are nine possibilities of relation; three products and six fractions.

Taking the physical formula for speed, we would have:

v=dt v=t/d v=d/t
t=dv t=d/v t=v/d
d=tv d=v/t d=t/v

(v=velocity or speed, d=distance travelled, t=time elapsed)

Checking the formula for speed

Now, focussing on the six fractions, check the mathematical formula for physical validity (all else remaining equal):

[Checking the different mathematical formulas for physical consistency; if either factor in a random mathematical product grows or diminishes physically, does the result do so as well?

Note that the two formulas found to be valid are interchangeable; and so are three pairs of products, bringing their number back to three.

So, magically (and relying on that, hopefully, no mistakes were made), this seems to have worked out fine for the formula for speed.

Checking the formula for electrical resistance

Let's check with the formula for electrical resistance (R=resistance, I=current, U=voltage):

[Checking the different mathematical formulas for physical consistency; if either factor in a random mathematical product grows or diminishes physically, does the result do so as well?

Again, depending on that no mistakes were made, it looks like the same result.

And so, everybody knows what's coming: Yes, of course, the hobby horse!

Checking the formula for entropy

Everybody do their thing… I have no idea…

[Checking the different mathematical formulas for physical consistency; if either factor in a random mathematical product grows or diminishes physically, does the result do so as well?

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