Unpossible / Impossible

Solution 1:

Ngram shows a prevalent usage of impossible vs unpossible also in the past centuries. It appears that unpossible has always been a less common variant:

Unpossible:

• (Etymology) from Middle English unpossible, equivalent to un- ‎(“not”) +‎ possible.

• (now rare, nonstandard) Impossible.

• 1526, William Tyndale, New Testament, British Library 2000, p. 119:

  • And this is the. vj. moneth to her, which was called barren, for with god shall nothinge be unpossible.
• 1621, Robert Burton, The Anatomy of Melancholy, New York Review of Books, 2001, p.280:
  • ’Tis a hard matter therefore to confine them, being they are so various and many, unpossible to apprehend all.

Impossible:

• (Etymology) from Old French impossible, from Latin impossibilis, from in- ‎(“not”) + possibilis ‎(“possible”), from possum ‎(“to be able”) + suffix -ibilis ‎(“-able”).

• Not possible; not able to be done or happen.

• 1300 Cursor M. 14761 It es bot foli al þi talking, And als an inpossibile [Gött. impossible] thing.

• 1340 Hampole Pr. Consc. 6281 Swa witty and myghty es he Þat na-thyng til hym impossibel may be.

Un-versus In- from Word Wide Words:

• In general, words take un- when they are of English (Germanic) origin and in- if they come from Latin. (The forms im-, il-, and ir- are variations on in-.) Apart from that, there’s really no good guide to which one you should choose. You’re just going to have to stick to learning them by rote.
• If it’s any consolation to you, the battle between in- and un- has been going on for centuries, with sometimes one form winning and sometimes the other, which suggests that the problem has been troubling English speakers for a very long time. As an example, for several centuries English had both inability and unability, but the latter disappeared in the eighteenth century for no very obvious reason.
• Another is familiar from the American Declaration of Independence: “We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable rights ...”; these days, it’s inalienable (it should always have been, by the rule, since alien comes from the Latin alienus, of or belonging to another person or place).

Solution 2:

Unpossible is not entirely verboten.

OED records it as "very common c1400–1660", but has citations from 1382 to the twentieth century and culminates with

2008 L. D. Brodsky Dine-rite 28 It's goddamn-to-hell next to unpossible To serve up taste-temptin' eats.

which is colloquial, but indicates that the word is used in some circumstances. Those circumstances would appear to be quoting a character's speech. ODO notes the word as "Now chiefly informal and regional".

Impossible is easily formed from the Latin impossibilis, and OED has citations from 1340.

It appears that the two words lived side-by-side for three hundred years before the Latinate impossible definitely gained the upper hand in the seventeenth century, when use of Latinate words became more popular. Because Google's Ngrams show relative frequency, and only go back to 1500 anyway (and there isn't an enormous amount of source material from that period), evidence from statistics may not be entirely reliable.