  We consider gedanken experiments to destroy an extremal or near-extremal BTZ
black hole by throwing matter into the horizon. These black holes are vacuum
solutions to (2+1)-dimensional gravity theories, and are asymptotically
$\mathrm{AdS}_3$. Provided the null energy condition for the falling matter, we
prove the following---(i) in a Mielke-Baekler model without ghost fields, when
torsion is present, an extremal BTZ black hole can be overspun and becomes a
naked conical singularity; (ii) in 3-dimensional Einstein gravity and chiral
gravity, which both live in torsionless limits of Mielke-Baekler model, an
extremal BTZ black hole cannot be overspun; and (iii) in both Einstein gravity
and chiral gravity, a near-extremal BTZ black hole cannot be overspun, leaving
the weak cosmic censorship preserved. To obtain these results, we follow the
analysis of Sorce and Wald on their gedanken experiments to destroy a
Kerr-Newman black hole, and calculate the second order corrections to the black
hole mass. Furthermore, Wald's type of gedanken experiment provides an
operational procedure of proving the third law of black hole mechanics. Through
the AdS/CFT correspondence, our results on BTZ black holes also indicate that a
third law of thermodynamics holds for the holographic conformal field theories
dual to 3-dimensional Einstein gravity and chiral gravity.
