We study a generalization of the Mermin-Peres magic square game to arbitrary rectangular dimensions. After exhibiting some general properties, these rectangular games are fully characterized in terms of their optimal win probabilities for quantum strategies. We find that for $m \times n$ rectangular games of dimensions $m,n \geq 3$ there are quantum strategies that win with certainty, while for dimensions $1 \times n$ quantum strategies do not outperform classical strategies. The final case of dimensions $2 \times n$ is richer, and we give upper and lower bounds that both outperform the classical strategies. Finally, we apply our findings to quantum certified randomness expansion to find the noise tolerance and rates for all magic rectangle games. To do this, we use our previous results to obtain the winning probability of games with a distinguished input for which the devices give a deterministic outcome, and follow the analysis of C. A. Miller and Y. Shi [SIAM J. Comput. 46, 1304 (2017)].

翻译：我们研究Mermin-Peres 魔术广场游戏的概括性,将其应用于任意的矩形维度。 在展示了某些一般特性之后, 这些矩形游戏的特征完全体现在它们最优的赢赢概率上。 我们发现,对于以美元计时的维度矩形游戏来说, $m,n\geq 3$, 有量子战略可以肯定地获胜, 而对于1美元计时量战略来说, 量子战略并不优于经典战略。 维度2美元的最后案例比较丰富, 我们给出了高于经典战略的上限和下限。 最后, 我们将我们的调查结果应用到数量认证的随机性扩展中, 以寻找所有魔术矩形游戏的噪音耐受度和速率。 为了做到这一点, 我们利用我们以前的结果, 来获得比赛的赢率, 其投入是卓越的, 设备给出了确定性结果, 并遵循C. A. Miller和Y. Shi. [SIAM J. Computurt. 46, 1304, (2017)] 的分析。