This paper details algorithmic innovations developed for the NVIDIA Nemotron Model Reasoning Challenge, specifically targeting bit manipulation puzzles where models must deduce hidden logical rules. To address the combinatorial explosion of bitwise operations and LLM hallucinations, the authors abandon arithmetic logic in favor of string similarity and structured search. The core contribution reframes logic-gate deduction as a base-selection task using minimal bit flips to isolate primitive transformations. A backtracking depth-first search process is formalized to test candidates, detect logical collisions, and perform robust error recovery. Additionally, the method employs bit tokenization and interactive reasoning supervised fine-tuning with dynamic masking to simulate oracle feedback. Evaluated on these puzzles, the approach achieved over 96% validation accuracy. This performance secured the highest result in the category and a seventh-place finish in the overall contest.