> there's a reason that the rules of algebra are what they are

But which algebra, and which rules ? There are infinitely many algebras ( eg. Algebra over the field of reals, Banach Algebra, relational algebra, boolean algebra, sigma algebra etc. ) The "rules" are really constructs you decide that apply to the elements of the space that conform to your algebra. So for example the reals are a field that have ordering, so you can talk about less than and greater than, but the complex numbers don't have an imposed order and you'd have to first define a norm to map them onto the reals. The AltDragonBox with its own inconsistent arbitrary rules will still have some algebraic encoding. Whether that's useful to you is debatable. Like in my algebra I could overload plus to mean multiply and square root to mean divide by 7 and add -3 and then try to figure out what exponentiation works out to. It would be interesting...maybe not useful, but its still an algebra. Maybe you won't have closure...the elements may not end up in a field or even in a semigroup...its a nice make-believe algebra.

Yes, that's a very impressive display of all the math you must surely know, but its completely misses the point. The rules of elementary algebra really are special, and it has to do with their correspondance to real things in real life. There's a reason mathematicians don't just enumerate all possible algebras and study them one by one.

I agree that it is important to recognize the limitations of this approach. As a high schooler (and even as an undergraduate in mathematics), you learn all sorts of formal systems. As a mathematics researcher, one of your charges is inventing interesting mathematical concepts to play around with. Obviously this is a bit much to ask out of DragonBox! And even more modestly, the ability to sit down with a problem and say, "Ah, but what really is going on here?" is a deep and difficult skill to impart. I don't know what these people were lacking, but if they didn't know how to move symbols around, I might have started there...

(BTW, an inconsistent set of rules would correspond to a version of DragonBox where there was a cheat code you could enter, and then easily solve every problem. So it would not "work just as well", and this would be pretty clear to a cheating gamer.)

OT: There's a great bit in Neal Stephenson's Anathem where these monk-like scientists are made to copy out subtly wrong mathematics and scientific proofs as a punishment.

AltDragonBox could be exactly that -- day and night cancel, except for symbols when its constellation is rising, in which case they divide, except for odd numbered Fridays in a leap year. Oh, and do it with numbers instead of day/night symbols.

Yes, there are loads of different forms of algebra and geometry. Sometimes people throw out what people though were fundamental rules and come up with new geometries (e.g. non-Euclidian geometry which turns out to be very helpful to describe relativity).

A lot of the time though, the other rules for alternative algebras produce very dull and boring algebras.