, 2024

Beginning at height 1 and moving up $6$ settings at a time, the desk can stop
at settings $7$, $13$, $19$, $25$, and $31$.

Beginning at height $31$ and moving
down 4 settings at a time, the desk can stop at settings $27$, $23$, $19$, $15$, $11$, $7$, and $3$.

The desk originally begins at an odd-numbered height, $1$.

Moving up $6$ settings at a time, the
desk can stop at only odd-numbered heights (since an even number added
to an odd number is odd).

Similarly, moving down $4$ settings
at a time, the desk can stop at only odd-numbered heights.

Thus, it is not possible for the desk to stop at an even-numbered
setting.

To this point, we have shown that the desk is able to stop at the
settings $$1,3,7,11,13,15,19,23,25,27,31,$$and is not
able to stop at even-numbered settings.

Next, we will show that it is possible for the desk to stop at the
remaining odd-numbered settings, $5$,
$9$, $17$, $21$, and $29$.

Since the desk can stop at setting $13$, then it can stop at settings $9$ and $5$ with one and two presses of the down
button, respectively.

Similarly, since the desk can stop at setting $25$, then it can stop at settings $21$ and $17$.

Finally, since the desk can stop at setting $23$, then one press of the up button will
take the desk to setting $29$.

The desk can stop at all odd-numbered settings from $1$ to $31$ inclusive, and thus is able to stop at
$16$ different settings.