Properties of ring counters and state utilization: Evaluate the statement: “In a ring counter, the modulus equals the number of flip-flops and, consequently, there are never any unused or invalid states.”

Introduction / Context:
Ring counters and Johnson counters are specialized shift-register counters used for simple sequencing. While a properly initialized ring counter cycles through a fixed set of states, it is important to understand what happens if the counter powers up in an unintended state and whether unused or invalid states exist.

Given Data / Assumptions:

• Ring counter: a single 1 (or 0) circulates through N flip-flops.
• Modulus under ideal initialization equals the number of flip-flops (N).
• Power-on state may be random without additional reset circuitry.

Concept / Approach:
Although the intended sequence has N valid states, the overall state space is 2^N. Any state that does not contain exactly one 1 (for a single-hot ring) is outside the intended sequence. Without a proper reset, the ring may lock into an invalid pattern (e.g., all zeros), which is unused and not part of the N-state loop. Therefore, the claim that there are “never” any unused or invalid states is false; design practice includes a reset to force entry into the valid cycle.

Step-by-Step Solution:

Verification / Alternative check:
Simulate a 4-bit ring counter with random initialization; only 4 of the 16 states belong to the ring sequence. Others are unreachable from the valid loop without reset or corrective feedback.

Why Other Options Are Wrong:

• Correct: Ignores the many unused states outside the single-hot set.
• Only true for Johnson counters: Johnson counters have 2N valid states but still have unused states overall.
• Only when initialized to all ones / Depends on synchronous reset: Initialization/reset affects entry into the valid loop but does not eliminate unused states from the state space.

Common Pitfalls:
Assuming power-on always yields a valid single-hot state; forgetting to include reset or self-correction logic; confusing ring and Johnson counter properties.

Final Answer:
Incorrect