Installing and Obstructing Heuristics: Learning Dynamics in Nim

The algorithmic capabilities of large language models may be driven by simple heuristics or spurious shortcuts. We study this phenomenon via Nim, a controlled natural language reasoning task in which the exact optimal strategy is known and the space of plausible shortcuts is explicit. In this setting, natural coarsenings of the target strategy correspond to coset structures of modular reduction, allowing us to directly observe when model performance plateaus at an intermediate heuristic during training. We show that these intermediate heuristics can be selectively implanted or circumvented through curriculum learning.