While AlphaZero-style reinforcement learning (RL) algorithms excel in various board games, in this paper we show that they face challenges on impartial games where players share pieces. We present a concrete example of a game - namely the children's game of Nim - and other impartial games that seem to be a stumbling block for AlphaZero-style and similar self-play reinforcement learning algorithms. Our work is built on the challenges posed by the intricacies of data distribution on the ability of neural networks to learn parity functions, exacerbated by the noisy labels issue. Our findings are consistent with recent studies showing that AlphaZero-style algorithms are vulnerable to adversarial attacks and adversarial perturbations, showing the difficulty of learning to master the games in all legal states. We show that Nim can be learned on small boards, but the learning progress of AlphaZero-style algorithms dramatically slows down when the board size increases. Intuitively, the difference between impartial games like Nim and partisan games like Chess and Go can be explained by the fact that if a small part of the board is covered for impartial games it is typically not possible to predict whether the position is won or lost as there is often zero correlation between the visible part of a partly blanked-out position and its correct evaluation. This situation starkly contrasts partisan games where a partly blanked-out board position typically provides abundant or at least non-trifle information about the value of the fully uncovered position.

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Endgame studies have long served as a tool for testing human creativity and intelligence. We find that they can serve as a tool for testing machine ability as well. Two of the leading chess engines, Stockfish and Leela Chess Zero (LCZero), employ significantly different methods during play. We use Plaskett's Puzzle, a famous endgame study from the late 1970s, to compare the two engines. Our experiments show that Stockfish outperforms LCZero on the puzzle. We examine the algorithmic differences between the engines and use our observations as a basis for carefully interpreting the test results. Drawing inspiration from how humans solve chess problems, we ask whether machines can possess a form of imagination. On the theoretical side, we describe how Bellman's equation may be applied to optimize the probability of winning. To conclude, we discuss the implications of our work on artificial intelligence (AI) and artificial general intelligence (AGI), suggesting possible avenues for future research.