Question 144
Set $M = \{(a, b, c, d) | a, b, c, d \in \{1, 2, 3, 4\} \text{ and } abcd \gt 1\}$. Find all values $(a_1, b_1)$ can assume if the sequence $(a_1, b_1, c_1, d_1)$, $(a_2, b_2, c_2, d_2)$, ..., $(a_{255}, b_{255}, c_{255}, d_{255})$ with $c_1 = d_1 = 1$ contains every element of $M$ exactly once and satisfies the condition
$$|a_{n+1} - a_n| + |b_{n+1} - b_n| + |c_{n+1} - c_n| + |d_{n+1} - d_n| = 1$$
for $1 \le n \lt 255$.
$$|a_{n+1} - a_n| + |b_{n+1} - b_n| + |c_{n+1} - c_n| + |d_{n+1} - d_n| = 1$$
for $1 \le n \lt 255$.