We want to design a synchronous counter that counts the sequence 0-1-0-2-0-3 and then repeats. The minimum number of J-K flip-flops required to implement this counter is _________ .

The required sequence is:

There are $6$ states in the repeating sequence cycle. Notice that the value $0$ appears three times, but is followed by different next values ($1$, $2$, and $3$). Therefore, to distinguish each occurrence of $0$ and form a deterministic finite state machine (counter), each step in the sequence of length $6$ must be assigned a unique state:

• $S_0 (0) \to S_1 (1) \to S_2 (0) \to S_3 (2) \to S_4 (0) \to S_5 (3) \to S_0 (0)$

The minimum number of distinct states required is $N = 6$.

The minimum number of flip-flops $k$ needed to represent $N$ states is given by:

With $3$ flip-flops, we can have up to $2^3 = 8$ distinct states, which is sufficient to encode the $6$ states. *(Note: An implementation using 4 flip-flops was also accepted in the official key range of 3 to 4).*