Large language models (LLMs) increasingly rely on chain-of-thought (CoT) prompting to solve mathematical and logical reasoning tasks. Yet, a central question remains: to what extent are these generated rationales \emph{faithful} to the underlying computations, rather than post-hoc narratives shaped by hints that function as answer shortcuts embedded in the prompt? Following prior work on hinted vs.\ unhinted prompting, we present a systematic study of CoT faithfulness under controlled hint manipulations. Our experimental design spans four datasets (AIME, GSM-Hard, MATH-500, UniADILR), two state-of-the-art models (GPT-4o and Gemini-2-Flash), and a structured set of hint conditions varying in correctness (correct and incorrect), presentation style (sycophancy and data leak), and complexity (raw answers, two-operator expressions, four-operator expressions). We evaluate both task accuracy and whether hints are explicitly acknowledged in the reasoning. Our results reveal three key findings. First, correct hints substantially improve accuracy, especially on harder benchmarks and logical reasoning, while incorrect hints sharply reduce accuracy in tasks with lower baseline competence. Second, acknowledgement of hints is highly uneven: equation-based hints are frequently referenced, whereas raw hints are often adopted silently, indicating that more complex hints push models toward verbalizing their reliance in the reasoning process. Third, presentation style matters: sycophancy prompts encourage overt acknowledgement, while leak-style prompts increase accuracy but promote hidden reliance. This may reflect RLHF-related effects, as sycophancy exploits the human-pleasing side and data leak triggers the self-censoring side. Together, these results demonstrate that LLM reasoning is systematically shaped by shortcuts in ways that obscure faithfulness.

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