Consider the given sequential circuit designed using D-Flip-flops. The circuit is initialized with some value (initial state). The number of distinct states the circuit will go through before returning back to the initial state is ________ . (Answer in integer)

The given circuit is a 4-bit Johnson counter (also known as a twisted ring counter or Moebius counter), where:

• $D_0 = \overline{Q_3}$

• $D_1 = Q_0$

• $D_2 = Q_1$

• $D_3 = Q_2$

For an $n$-bit Johnson counter, the main cycle consists of $2n$ distinct states. Here, $n = 4$, so the cycle length is:

For example, starting with the initial state $(Q_0, Q_1, Q_2, Q_3) = 0000$:

1. $0000$ (initial state)

2. $1000$

3. $1100$

4. $1110$

5. $1111$

6. $0111$

7. $0011$

8. $0001$

9. $0000$ (returns to initial state)

Depending on the interpretation of the phrase *"distinct states the circuit will go through before returning back to the initial state"*:

• If counting the intermediate distinct states visited after the start and strictly before the return state: 7 states.

• If counting all distinct states visited in the cycle (including the initial state): 8 states.

Hence, both 7 and 8 are accepted according to the official key.