How can we force the failure of $\square(\kappa)$ at an inaccessible $\kappa$, where 
$\square(\kappa)$ is defined as follows: There is a sequence $(C\_i:i< \kappa)$ such that:


(1) $C\_{i+1} = \{i\}$ and $C\_i$ is closed and cofinal in $i$ if $i$ is a limit
ordinal.


(2) If $i$ is a limit point of $C\_j$, then $C\_i = C\_j \cap i$.


(3) There is no club $C$ (a subset of $\kappa$) such that for all limit points $i$ in
$C$ the equality $C\_i= C \cap i$ holds.


**Update.** By a recent result of Magidor and Vaananen ( On Löwenheim-Skolem-Tarski numbers for extensions of first order logic), it is consistent, relative to the existence of a supercompact cardinal, that $\square(\kappa)$ fails at the least strongly inaccessible cardinal. 


Is it possible to reduce the consistency strength to a weakly compact cardinal or even a measurable cardinal?



failure of $\square(\kappa)$ at an inaccessible $\kappa$
The consistency strength of the failure of $\square(\kappa)$ for non-weakly Mahlo inaccessible cardinal $\kappa$ (in particular, for the first strongly inaccessible) is higher than weakly compact. For example it implies that $0^\#$ exists.
<http://www.jstor.org/stable/27590333>
We know that global square holds in $L$, i.e. there is a sequence $\langle C\_\alpha | \alpha \text{ is singular limit ordinal} \rangle$ with the properties: $\forall \alpha,\, \text{otp } C\_\alpha < \alpha$ and for every accumulation point $\beta$ of $C\_\alpha$, $C\_\alpha \cap \beta = C\_\beta$. 


In $V$, there is a club $D\subset \kappa$ through the singular cardinals below $\kappa$. If $0^\#$ doesn't exists, the covering theorem will imply that those cardinals are singular in $L$ as well, and therefore $C\_\alpha$ is defined for every $\alpha \in D$.


Now we can define a $\square(\kappa)$ sequence, $\langle E\_\alpha | \alpha < \kappa \rangle$ in $V$ as follow:


* For $\alpha \in \text{acc } D$, $E\_\alpha = C\_\alpha \cap D$ (unless this set is bounded below $\alpha$. In this case, $\text{cf }\alpha = \omega$ and we can choose $E\_\alpha$ to be arbitrary $\omega$ sequence cofinal in $\alpha$).
* For $\alpha \notin \text{acc }D$, $E\_\alpha = \alpha \setminus (\max D\cap \alpha)$


It's easy to see that $\langle E\_\alpha | \alpha < \kappa\rangle$ is coherent (since the global square is coherent) and thread-less (since every thread meets $D$ club many times, but $\text{otp }E\_\alpha < \alpha$ for every $\alpha \in D$).


**Edit**: As Mohammad noted, we can get better lower bound by using more appropriate inner model. We need to violate global square on the singular cardinals. By [$\square$ On the singular cardinals](http://www.jstor.org/stable/27590333) (Theorem 1), we need at least many measurable cardinals of uncountable Mitchell order in order to get a model without global square on the singular cardinals. 


