  We consider the random number partitioning problem (\texttt{NPP}): given a
list $X\sim \mathcal{N}(0,I_n)$ of numbers, find a partition
$\sigma\in\{-1,1\}^n$ with a small objective value
$H(\sigma)=\frac{1}{\sqrt{n}}\left|\langle \sigma,X\rangle\right|$. The
\texttt{NPP} is widely studied in computer science; it is also closely related
to the design of randomized controlled trials. In this paper, we propose a
planted version of the \texttt{NPP}: fix a $\sigma^*$ and generate $X\sim
\mathcal{N}(0,I_n)$ conditional on $H(\sigma^*)\le 3^{-n}$. The \texttt{NPP}
and its planted counterpart are statistically distinguishable as the smallest
objective value under the former is $\Theta(\sqrt{n}2^{-n})$ w.h.p.
  Our first focus is on the values of $H(\sigma)$. We show that, perhaps
surprisingly, planting does not induce partitions with an objective value
substantially smaller than $2^{-n}$: $\min_{\sigma \ne \pm \sigma^*}H(\sigma) =
\widetilde{\Theta}(2^{-n})$ w.h.p. Furthermore, we completely characterize the
smallest $H(\sigma)$ achieved at any fixed distance from $\sigma^*$. Our second
focus is on the algorithmic problem of efficiently finding a partition
$\sigma$, not necessarily equal to $\pm\sigma^*$, with a small $H(\sigma)$. We
show that planted \texttt{NPP} exhibits an intricate geometrical property known
as the multi Overlap Gap Property ($m$-OGP) for values $2^{-\Theta(n)}$. We
then leverage the $m$-OGP to show that stable algorithms satisfying a certain
anti-concentration property fail to find a $\sigma$ with
$H(\sigma)=2^{-\Theta(n)}$.
  Our results are the first instance of the $m$-OGP being established and
leveraged to rule out stable algorithms for a planted model. More importantly,
they show that the $m$-OGP framework can also apply to planted models, if the
algorithmic goal is to return a solution with a small objective value.
