In digital counters, a ring counter advances a single 1 (one-hot) through its flip-flops. For a modulus-12 (mod-12) ring counter, what is the minimum number of flip-flops required to realize exactly 12 unique states?

Introduction / Context:
Ring counters are a special class of shift-register counters widely used in timing, control sequencing, and state machine design. They employ a one-hot code in which exactly one flip-flop is set at any time and the 1 circulates around each clock pulse. This question checks whether you understand how the modulus (the total number of distinct states) relates directly to the number of flip-flops in a ring counter.

Given Data / Assumptions:

• Counter type: ring counter (one-hot).
• Desired modulus (number of states): 12.
• Each flip-flop stores a single bit and participates equally in the rotation of the 1.

Concept / Approach:
In a ring counter, the number of valid states equals the number of flip-flops when exactly one flip-flop is high and the rest are low. Therefore, a mod-N ring counter requires N flip-flops. This is unlike binary ripple or synchronous counters where the number of states is 2^n for n flip-flops.

Step-by-Step Solution:
1) Define modulus requirement: we need 12 distinct circulating one-hot states.2) In a ring counter, each flip-flop position corresponds to one unique state.3) Therefore, number of flip-flops required = number of states = 12.4) Minimum flip-flop count that exactly meets mod-12 = 12 flip-flops.

Verification / Alternative check:
If fewer than 12 flip-flops were used (e.g., 10), only 10 distinct one-hot positions would exist, so the modulus would be 10, not 12. Conversely, using more than 12 flip-flops would create a larger modulus than required.

Why Other Options Are Wrong:
10 flip-flops: gives mod-10, not mod-12.6 flip-flops: gives mod-6, not mod-12.2 flip-flops: gives mod-2, trivial and incorrect here.8 flip-flops: gives mod-8, still not 12.

Common Pitfalls:
Confusing ring counters with binary counters. Binary counters use 2^n states; ring counters use n states. Another pitfall is assuming Johnson (twisted-ring) logic, which yields 2n states; this problem specifies a plain ring counter, not a Johnson counter.

Final Answer:
12 flip-flops