The letters L, M, N, O, P, Q, R, S and T in their order are substituted by nine integers $1$ to $9$ but not in that order. $4$ is assigned to P. The difference between P and T is $5$. The difference between N and T is $3$. What is the integer assigned to N?

Let the integers assigned to the letters be their values from the set $\{1,2,3,4,5,6,7,8,9\}$.
Given:
$P=4$

The absolute difference between P and T is $5$:

$$|P-T|=5⟹|4-T|=5$$

Since $T$ must be between $1$ and $9$, we have:
$T=4+5=9$ (as $4-5=-1$ is out of range).

The absolute difference between N and T is $3$:

$$|N-T|=3⟹|N-9|=3$$

Since $N$ must be a valid integer between $1$ and $9$:
$N=9-3=6$ (as $9+3=12$ is out of range).

Therefore, the integer assigned to N is $6$.