Let
$$
D(\lambda)=\#\{\text{odd hook lengths of }\lambda\}
-\#\{\text{even hook lengths of }\lambda\}.
$$
A direct comparison of the affected bead-gap pairs shows that removing a rim 2-hook preserves $D$: the pairs whose parities change cancel in odd-even pairs. Repeating this removal gives
$$
D(\lambda)=D(C_2(\lambda)).
$$
Every <core of a partition>[2-core] is a staircase
$$
(r,r-1,\ldots,1).
$$
All hook lengths in this staircase are odd, and it has $1+2+\cdots+r=r(r+1)/2$ cells. Hence the <odd-minus-even hook count of a partition> is
$$
\boxed{D(\lambda)=\binom{r+1}{2}}.
$$
Thus the requested integer is $m=r+1=\ell(C_2(\lambda))+1$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 3 / b / Solution

Removing a rim hook of length two preserves the number of odd hook lengths minus the number of even hook lengths. If the 2-core is the staircase $(r,r-1,\ldots,1)$, every one of its $r(r+1)/2$ hooks is odd. Hence the difference for the original partition is
$$
\binom{r+1}{2}.
$$


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