[wildfyr]

rlabomba yep, that is exactly what I had in mind! I'm fairly sure that was the test answer expected. Although, once again, Don't capitalize the "R" in "Br". It implies you are using a weird boronic ester, not diatomic bromine. Just to be specific, I would say the first Br₂ is in the presence of light, the second one isnt, or uses NBS as a bromine source.

Organicdan,
I had this discussion with a professor a long time ago, the reaction simply doesn't work in real life. OH^-'s basic properties simply overcome its nucleophilicity, even under cold temps. I challenge you to find a scholarly source where OH^- will displace an alkyl halide directly, even a primary one. I used scifinder for this cyclohexane substitution and was unable to find an example. This is a case where paper chemistry should bow to what we observe in the real world.

I did find a couple examples where a primary alkyl halide in mixed aqueous/organic solutions could be cooked to give an alcohol (like this http://pubs.acs.org/doi/abs/10.1021/jo00156a045), but that is water, not NaOH. Another exception is that some alkyl iodides will slowly hydrolyze in neutral water.

[Enthalpy]

How long would cyclohexyne exist?

I have no opinion whatsoever, just saw that Wiki tells
"only exist as transient reaction intermediates or as ligands coordinating to a metal center"
https://en.wikipedia.org/wiki/Cycloalkyne
because of the strain.

[wildfyr]

Enthalpy,
I dont think OP is proposing to make this on the bench. Its from a test.

[kriggy]

does it matter? Why would you put such a question on a test? While one might miss some examples that wont exist in real life, this one seems unnecessary because you can just make it linear alkyne and be fine..
Not to mention, the double elimination might be tricky as well even for larger rings (8 membered)

[OrganicDan96]

does it matter? Why would you put such a question on a test? While one might miss some examples that wont exist in real life, this one seems unnecessary because you can just make it linear alkyne and be fine..
Not to mention, the double elimination might be tricky as well even for larger rings (8 membered)

and why would you start with such a molecule that contains no functional groups?

[kriggy]

Why not?
As a TA, I would say that making molecules from stupid starting materials that you would never use in a lab is fine as a learning tool but I would avoid compounds that are not reasonably stable (such as the cyclohexyne)

[Enthalpy]

Supposedly, the professor will be happy with a synthesis applicable to linear alkynes.

Wiki gives syntheses for the cycloalkynes
https://en.wikipedia.org/wiki/Cycloalkyne#Synthesis
and they differ from linear alkynes, at least to my untrained eye. Possibly beyond the scope of the course.

If I grab it properly, cyclohexyne would only serve in situ and immediately for subsequent reaction steps.

[Babcock_Hall]

When one has dibromocyclohexane as the substrate for an elimination, what would prevent its becoming cyclohexadiene?

[wildfyr]

Very good point Babcock. This may in fact be more likely since allylic protons are 2 pKa units lower than alkenic ones. Perhaps we NEED the gem dibromo to force the alkyne. I doubt it will double eliminate in two directions to give an internal ketene from the gem dibromo cyclohexane.