We denote an arbitrary permutation of the integers $1$, $2$, $\ldots$, $n$ by $a_1$, $a_2$, $\ldots$, $a_n$.  Let $f(n)$ denote the number of these permutations such that:

(1) $a_1 = 1$;

(2):$|a_i - a_{i+1}| \leq 2$, $i = 1, \ldots, n - 1$.

Determine whether $f(1996)$ is divisible by 3.
To solve the problem, we need to determine whether \( f(1996) \) is divisible by 3. We start by analyzing the recurrence relation and the properties of \( f(n) \).

1. **Define the recurrence relation:**
   Given the conditions:
   - \( a_1 = 1 \)
   - \( |a_i - a_{i+1}| \leq 2 \) for \( i = 1, \ldots, n - 1 \)

   We need to find a recurrence relation for \( f(n) \). From the problem, we have:
   \[
   f(n+3) = f(n) + f(n+2) + 1 \quad \forall n > 1
   \]

2. **Analyze the recurrence relation modulo 3:**
   We need to determine the values of \( f(n) \mod 3 \) for different \( n \). Let's start by computing the first few values of \( f(n) \) to identify a pattern.

   - \( f(1) = 1 \)
   - \( f(2) = 1 \)
   - \( f(3) = 2 \)
   - \( f(4) = f(1) + f(3) + 1 = 1 + 2 + 1 = 4 \equiv 1 \mod 3 \)
   - \( f(5) = f(2) + f(4) + 1 = 1 + 1 + 1 = 3 \equiv 0 \mod 3 \)
   - \( f(6) = f(3) + f(5) + 1 = 2 + 0 + 1 = 3 \equiv 0 \mod 3 \)
   - \( f(7) = f(4) + f(6) + 1 = 1 + 0 + 1 = 2 \equiv 2 \mod 3 \)
   - \( f(8) = f(5) + f(7) + 1 = 0 + 2 + 1 = 3 \equiv 0 \mod 3 \)

3. **Identify the pattern:**
   From the above calculations, we observe the following pattern:
   \[
   \begin{align*}
   f(1) &\equiv 1 \mod 3 \\
   f(2) &\equiv 1 \mod 3 \\
   f(3) &\equiv 2 \mod 3 \\
   f(4) &\equiv 1 \mod 3 \\
   f(5) &\equiv 0 \mod 3 \\
   f(6) &\equiv 0 \mod 3 \\
   f(7) &\equiv 2 \mod 3 \\
   f(8) &\equiv 0 \mod 3 \\
   \end{align*}
   \]

   This pattern repeats every 8 terms. Specifically:
   \[
   \begin{align*}
   f(8k-1) &\equiv 2 \mod 3 \\
   f(8k) &\equiv 0 \mod 3 \\
   f(8k+1) &\equiv 1 \mod 3 \\
   f(8k+2) &\equiv 1 \mod 3 \\
   f(8k+3) &\equiv 2 \mod 3 \\
   f(8k+4) &\equiv 1 \mod 3 \\
   f(8k+5) &\equiv 0 \mod 3 \\
   f(8k+6) &\equiv 0 \mod 3 \\
   f(8k+7) &\equiv 2 \mod 3 \\
   \end{align*}
   \]

4. **Determine \( f(1996) \mod 3 \):**
   Since \( 1996 = 8 \times 249 + 4 \), we have:
   \[
   f(1996) \equiv f(8 \times 249 + 4) \equiv f(4) \equiv 1 \mod 3
   \]

Conclusion:
\[
f(1996) \equiv 1 \mod 3
\]

The final answer is \( \boxed{ 1 \mod 3 } \)