Is "architect" a verb and a noun?

Traditionally "architect" is a noun only, but it is increasingly common to hear it used as a verb, though usually in business or technical situations where jargon is very common. Since the language is largely defined by its usage, and (as Robusto notes) nouns often become verbs and vice versa, it's hard to say that it's wrong - but many consider it poor style, and in formal writing (especially if intended for an audience unfamiliar with business-speak) it might be better to choose an alternative that conveys the required meeting - perhaps "design", or "build", or even "think of" or "solve" in the specific example you gave :-)

Merriam-Webster’s Third International lists “architect” as a verb. So does the Oxford English Dictionary (probably the most authoritative and prestigious dictionary ever compiled for any human language), with citations going back to the early eighteen hundreds:

To design (a building). Also transf. and fig.

1818 Keats Let. July (1958) I. 350 This was architected thus By the great Oceanus. [But see architecture v.]

1890 Harper's Mag. Apr. 809/2 We would not give being the author of one of Mr. Aldrich's beautiful sonnets to be the author of many ‘Wyndham Towers’, however skilfully architected.

1913 W. Raleigh Some Authors (1923) 3 He has come out of the prison-house of theological system, nobly and grimly architected.

Personally, as a data architect, I find this a useful verb, since I use it to include analysis and integration, as well as design. It also encompasses the standards for deliverables associated with the architecture process.

Those who do not find this term useful are not required to use it. But it is inappropriate to sneer at those who do.