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My friend Della and I constantly try to invent things. We're currently working on a patent for a wondrous machine which we both designed: the magnificent Bubblizer!

What does it do? Well... right now, not much. We're still working out all the kinks of this contraption, since it's still just a prototype. I'll tell this much — and really, this is a wondrous thing: it can turn anyone... into pure NEON! (Cue mad-scientist music and lightning effects)

Yes, yes, I know; it's astounding! Really, it's the greatest! Some of the best work Della and I have made! And you know what? It gets better! You waiting?... It's perfectly* reversible! Yes, you heard right! The best part: completely pain-free! Wanna try?...

You sure?...

Ah, maybe another time...

Perhaps it's for the best anyways; you see, we encountered a bug during our latest software update. All I tried to do was add a... small feature... I thought it would make the Bubblizer the next-best thing with all the young kids, but Della said no, and I-... well, see for yourself:

I wanted the Bubblizer to handle some basic math, but for some reason the Bubbilizer is giving me some rather strange results. For example:

  • I plug in 2 + 6, but I get 24,076,222
  • I plug in 1 × 10, but I get 746,375,756,370
  • I plug in 5 - 4, but I get -2,096,046
  • I plug in 9 ÷ 0, but I get 0.28323633188

Maybe it is a feature, but Della keeps saying it's just my inputs. Certainly that can't be right... right? Maybe you can figure out what went wrong?

The answer to this question is a simple one: explain what's going on in each input and output to figure out how my– er, our machine works. And I do mean each one (including the secret one).

*Of course, "perfectly reversible" is relative; like any reasonable scientific endeavor, the process must obey the laws of thermodynamics. While it's possible to revert one back to their initial state, the reverse process does add... an usual appendage. Usually an extra nose or another nail on their toes... otherwise, perfectly normal and entirely painless!

WARNING: this claim has not been verified by the FDA.

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The Bubblizer does indeed

make any ONE into NEON

by

turning the word inside out: start in the middle and then read to the end, then start at the beginning and then read to the middle.

If you do this twice

then for a word of even length you get the original back unaltered; for a word of odd length, though, you get an extra appendage. E.g., ABCDE -> CDEABC -> ABCCDE. The appendage is in the middle rather than at one end, though, contrary to the description in the question.

As Della says, the arithmetic results

are strange only because of the inputs. The numbers have been written down in English and then had the Bubblizer procedure applied to them before converting back to numbers via A1Z26 and concatenation, and then doing the arithmetic. So e.g. ONE * TEN -> NEON * ENTE -> 1451514 * 514205, which does indeed yield the stated result.

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  • $\begingroup$ Indeed, this is correct! I wonder what gave it away? Perhaps I made it too easy and should have only used two inputs; still, good job. Della won't be too happy that I gave away our patent secrets so easily, though.... $\endgroup$
    – Omicron Zed
    Feb 24, 2020 at 0:00
  • $\begingroup$ Just to note: the whole "extra appendage" hint was meant to be a bit of a quasi-red-herring. Other than that, I actually used a slightly different algorithm to switch words back-and-forth, although it comes to the same ending and behaves quite similar and your answer captures all the requirements I needed. $\endgroup$
    – Omicron Zed
    Feb 24, 2020 at 0:24
  • $\begingroup$ I don't think you made it too easy. The first observation that pointed me in the right direction was that rot13(va nyy gur nevguzrgvp pnfrf, gur gjb bcrenaqf jrer gur fnzr yratgu). The rot13("nalbar gb arba") thing was also fairly obviously significant, which was helpful later. $\endgroup$
    – Gareth McCaughan ♦
    Feb 24, 2020 at 0:58
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    $\begingroup$ The rot13(zhygvcyvpngvba naq qvivfvba) gave much more information than the rot13(nqqvgvba naq fhogenpgvba) and I probably wouldn't have got it with only the latter. With the former, I could rot13(gel nyy snpgbevmngvbaf bs gur ovt ahzore vagb gjb cnegf bs pbzcnenoyr zntavghqr naq gel inevbhf jnlf bs vagrecergvat gur gjb snpgbef, naq ybbx sbe senpgvbaf rdhny gb gur tvira qvivfvba erfhyg naq ntnva gel inevbhf jnlf bs vagrecergvat gur ahzrengbef naq qrabzvangbef), whereas rot13(jvgu nqqvgvba naq fhogenpgvba vg'f zhpu uneqre gb tb sebz gur bhgchg gb gur vachgf). $\endgroup$
    – Gareth McCaughan ♦
    Feb 24, 2020 at 1:01

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