So, there are 20 ways to choose positions for the 3 A’s such that no two are adjacent. - Abbey Badges

Understanding the Context

2. Binary Selection with Gaps

Use a binary pattern where A’s occupy positions marked by 1 separated by at least one 0. For three A’s, valid binary-like selections include positions 1, 3, 5 or 2, 5, 7.

3. Reverse Intervals

Choose positions linked by consistent offsets, such as 1, 4, 7 — spacing 3 apart ensures no adjacency. This spreads out the A’s evenly.

4. Field-Based Sampling

If positions relate to a fixed set (e.g., zones in a layout), sample non-consecutive zones such as 2, 5, and 8 from a list of 10 zones.

5. Randomized Selections with Validation

Randomly pick 3 positions, then validate if any two are adjacent; reroll until all are non-adjacent. Combines speed with accuracy.

Key Insights

6. Gap-Ensuring Algorithm

Apply a mathematical approach where the first A is placed freely, the second at least one gap away, and the third at least one gap from the second—ensuring minimum two separators.

7. Maximum Separation Strategy

Maximize distance by selecting positions as far apart as possible, such as 1, 4, and 8, minimizing clustering and maximizing coverage.

8. Symmetric Spacing

Use symmetry by mirroring positions around a center, e.g., picking 1, 4, and 7 (symmetric with respect to center 4) for balanced distribution.

9. Periodic Placement Using Modulo

Assign positions using modular arithmetic—e.g., select all positions ≡ 1 mod 4 (1, 5, 9)—ensuring consistent gaps.

10. Algorithm-Assisted Selection

Leverage combinatorial algorithms that generate non-adjacent triplets without overlap efficiently.

Final Thoughts

11. Exclusion and Filtering

Start with all positions, then iteratively exclude positions adjacent to already selected A’s, filtering valid options in real-time.

12. Visual Placement Board

Use cards or layout boards to physically position markers—for 3 A’s, space them visibly with at least one empty space between each.

13. Weighted Probability Based on Gap Length

Assign higher selection likelihood to positions with longer gaps, ensuring wider spacing naturally.

14. Recursive Backtracking for Optimal Choices

For complex constraint sets, a recursive method tries all combinations filtration and backtracks to find valid triplets.

15. Circular Layout Consideration

If positions wrap (like circular layouts), ensure first and last are not adjacent—select 1, 4, 7 without wrapping to 10 adjacent to 1.

16. Sliding Window Technique

Apply a sliding window of fixed size (≥3) across the position array, capturing only triplets where no two selected spots are allocated in adjacent windows.