Recent trends in mathematics education emphasize discovery learning over drill. This has proven to be a bad idea in some cases, for the simple reason that practice is required to learn basic arithmetic skills. Drills in arithmetic skills can be made interesting through gamification. This study proposes a family of puzzles that gamify arithmetic practice. The puzzles are designed with an evolutionary algorithm forming an instance of automatic content generation. Two methods of evolutionary puzzle design are presented and discussed. The first method used transformed the problem into an almost trivial optimization. The second algorithm was designed to avoid the flaws of the first and produced a huge variety of puzzles. A hardness measure, based on the difficulty experienced by the evolutionary puzzle generator, is employed. The hardness measure is tested on a large collection of puzzles produced with the evolutionary automatic content generation system. An initial assumption, that all the pieces in the puzzle must be used to achieve a maximum score, was shown to be incorrect in puzzles located via automatic search. Two classes of puzzle are defined: those where the optimal solution uses all pieces and those where the optimal solution fails to use at least one piece. The latter sort of puzzle were found to be far more common in the search space.

中文翻译：

---

用于数学实践的多米诺拼图

数学教育的最新趋势强调发现学习而非钻研。事实证明，在某些情况下，这是个坏主意，原因很简单，需要实践才能学习基本的算术技能。通过游戏化可以使演算技巧变得有趣。这项研究提出了一系列将算术实践游戏化的难题。拼图采用进化算法设计，形成了自动内容生成的一个实例。提出并讨论了两种进化拼图设计方法。使用的第一种方法将问题转化为几乎不重要的优化。第二种算法的设计避免了第一种算法的缺陷，并产生了各种各样的难题。基于进化难题生成器所经历的难度，采用硬度度量。硬度测量在使用进化式自动内容生成系统生成的大量拼图上进行了测试。最初的假设表明拼图中的所有碎片都必须使用以达到最大分数，这在通过自动搜索定位的拼图中被证明是不正确的。定义了两类难题：那些最佳解决方案使用所有零件的难题，以及那些最佳解决方案无法使用至少一件零件的难题。发现后一种难题在搜索领域更为普遍。那些使用最佳解决方案使用所有部件的产品，以及那些使用最佳解决方案无法使用至少一件产品的产品。发现后一种难题在搜索领域更为普遍。那些使用最佳解决方案使用所有部件的产品，以及那些使用最佳解决方案无法使用至少一件产品的产品。在搜索空间中发现后一种难题更为普遍。