AI 中文总结
该研究提出一个等价于整数因式分解的单人棋盘博弈，并利用AlphaZero风格的蒙特卡洛树搜索在受限搜索空间中探索神经前瞻的极限，以求解该博弈。
AI 中文摘要
我们研究了一个在$N\times N$棋盘上进行的单人棋子博弈，其中棋子沿对角线滑动或复制到相邻位置以形成组合矩形$R\times S$。一个守恒的整数权重$W'$和一个严格单调不变量保证了$O(N^2)$长度的解，使该博弈属于$\mathsf{NP}$。我们证明，达到最终位置会将这个$2N$位的$W'$分解为两个$N$位的因子$V,M < 2^N$，它们编码了矩形的行和列。因此，对于平衡半素数目标，求解该博弈等价于整数因式分解。然而，如果目标矩形已知，则解简化为两个多项式时间步骤：一个强制向下筹码流和利用Cohn定理的$0/1$多项式因式分解。因此，博弈的全部难度被隔离到初始的数论分割中。提供因子汉明重量作为承诺保留了这种渐近难度，但限制了目标搜索空间。我们利用这个受限空间，采用学习策略/价值网络和AlphaZero风格的蒙特卡洛树搜索，在因式分解等价环境中实证探索神经前瞻的极限。
英文摘要
We study a one-player token game on an $N\times N$ board where tokens slide along diagonals or duplicate onto neighbouring ones to form a combinatorial rectangle $R\times S$. A conserved integer weight $W'$ and a strict monovariant guarantee $O(N^2)$-length solutions, placing the game in $\mathsf{NP}$. We prove that reaching a final position factors this $2N$-bit $W'$ into two $N$-bit factors $V,M < 2^N$ that encode the rectangle's rows and columns. Consequently, solving the game for a balanced-semiprime target is equivalent to integer factoring. However, if the target rectangle is known, the solution reduces to two polynomial-time steps: a forced downward chip-flow and a $0/1$-polynomial factorisation leveraging Cohn's theorem. The game's entire difficulty is thus isolated to the initial number-theoretic split. Supplying the popcounts of the factors as a promise preserves this asymptotic hardness but bounds the target search space. We exploit this constrained space using a learned policy/value network and an AlphaZero-style Monte-Carlo tree search, empirically probing the limits of neural look-ahead on a factoring-equivalent environment.
