  For a Gray code in the scheme of rank modulation for flash memories, the
codewords are permutations and two consecutive codewords are obtained using a
push-to-the-top operation. We consider snake-in-the-box codes under Kendall's
$\tau$-metric, which is a Gray code capable of detecting one Kendall's
$\tau$-error. We answer two open problems posed by Horovitz and Etzion.
Firstly, we prove the validity of a construction given by them, resulting in a
snake of size $M_{2n+1}=\frac{(2n+1)!}{2}-2n+1$. Secondly, we come up with a
different construction aiming at a longer snake of size
$M_{2n+1}=\frac{(2n+1)!}{2}-2n+3$. The construction is applied successfully to
$S_7$.
