The Logic: The "Drowning Man" Scenario
The sentence describes a specific psychological state: urgency combined with a lack of obvious solutions.
- The Context: When we are desperate for an answer, our standards for what constitutes a "good" suggestion drop.
- The Pivot: The word "however" functions here as a concession. It suggests that even if the suggestion is weak, unproven, or shaky, we will still "welcome" it because we have no other options.
- The Goal: We need two words that describe a suggestion that is weak, non-committal, or lacking a solid foundation.

The Correct Answers:
C. tentative
F. flimsy
- Why they work: * Tentative means hesitant, unconfirmed, or not fully worked out.
- Flimsy means weak, superficial, or lacking substance.
- In an urgent crisis, even a tentative (unproven) or flimsy (weak) idea is better than no idea at all. Both words convey the "low quality" of the suggestion that we are nonetheless willing to accept.

Why the others fail:
- A. unusual and D. outrageous: While we might welcome an unusual or outrageous idea, these two words are not synonyms in this context. An outrageous idea is bold/shocking, while an unusual one is simply rare.
- B. hackneyed: This means overused or cliché. While a cliché might be boring, it is usually a "solid" (if unoriginal) solution, which doesn't fit the idea of a suggestion that is barely "throwing light" on a dilemma.
- E. illuminating: This is a "trap" word. The sentence says the suggestion might "throw light" (illuminate), but using "however illuminating" would imply we are reluctant to accept helpful ideas, which makes no sense.

Final Completed Sentence
When faced with an urgent problem for which there is no immediately obvious solution, we tend to welcome any suggestion, however tentative / flimsy, that might throw light on the dilemma.