I tried again like this, supposing we know the parity case, for example odd on first day
By checking (1,3),(4,6),(7,9),..., we can check a range of (3k-1) in k days. Now I construct a sequence of numbers ai, such that
an >= 3*sumi<nai
we can easily verify that ai = 4i qualifies.
The strategy now becomes,
in the first a1=4 days, we check (-5,-3),(-2,0),(1,3),(4,6), which is a range of -5 -> 6
in the next a2 = 16 days, we check (-23, -21),(-20,-18),...(-2,0),(1,3),..,(22,24), which is a range of -23 -> 24
etc.
Supposing groundhog is at P, while