We are given three strings, $s$, $t$ and $p$. We will denote the length of a string by vertical bars, thus $|s|$ is the length of $s$ and so on. If we insert $t$ into $s$ at position $k$, where $0 \le k \le |s|$, the result is a new string consisting of the first $k$ characters of $s$, followed by the entirety of $t$, and finally followed by the remaining $|s| - k$ characters of $s$. We would like to select $k$ so that the resulting new string will contain the largest possible number of occurrences of $p$ as a substring.

Thus, for example, inserting $t = \texttt{aba}$ into $s = \texttt{ab}$ at position $k = 0$ results in the string $\texttt{abaab}$; at $k = 1$, in the string $\texttt{aabab}$; and at $k = 2$, in the string $\texttt{ababa}$. If we are interested in occurrences of $p = \texttt{aba}$, then the best position to insert $t$ into $s$ is $k = 2$, where we get two occurrences: $\texttt{ababa}$ and $\texttt{ababa}$ (as this example shows, occurrences of $p$ are allowed to overlap). If, on the other hand, we were interested in occurrences of $p = \texttt{aa}$, then the best choices of $k$ would be $k = 0$ and $k = 1$, which result in one occurrence of $p$, whereas $k = 2$ results in 0 occurrences of $p$.