In 2022, Olivier Longuet, a French mathematics teacher, created a game called the \textit{calissons puzzle}. Given a triangular grid in a hexagon and some given edges of the grid, the problem is to find a calisson tiling such that no input edge is overlapped and calissons adjacent to an input edge have different orientations. We extend the puzzle to regions $R$ that are not necessarily hexagonal. The first interesting property of this puzzle is that, unlike the usual calisson or domino problems, it is solved neither by a maximal matching algorithm, nor by Thurston's algorithm. This raises the question of its complexity. We prove that if the region $R$ is finite and simply connected, then the puzzle can be solved by an algorithm that we call the \textit{advancing surface algorithm} and whose complexity is $O(|\partial R|^3)$ where $\partial R|$ is the size of the boundary of the region $R$. In the case where the region is the entire infinite triangular grid, we prove that the existence of a solution can be solved with an algorithm of complexity $O(|X|^3)$ where $X$ is the set of input edges. To prove these theorems, we revisit William Thurston's results on the calisson tilability of a region $R$. The solutions involve equivalence between calisson tilings, stepped surfaces and certain DAG cuts that avoid passing through a set of edges that we call \textit{unbreakable}. It allows us to generalize Thurston's theorem characterizing tilable regions by rewriting it in terms of descending paths or absorbing cycles. Thurston's algorithm appears as a distance calculation algorithm following Dijkstra's paradigm. The introduction of a set $X$ of interior edges introduces negative weights that force a Bellman-Ford strategy to be preferred. These results extend Thurston's legacy by using computer science structures and algorithms.

翻译：2022年，法国数学教师奥利维耶·隆盖创造了一款名为"卡里松拼图"的游戏。给定六边形中的三角形网格及若干指定边，问题在于寻找一种卡里松铺砌方案，使得输入边不被覆盖，且与输入边相邻的卡里松块朝向不同。我们将该拼图推广至非六边形区域$R$。该拼图的第一个有趣特性是：与常规卡里松或多米诺骨牌问题不同，它既不能通过最大匹配算法求解，也无法通过瑟斯顿算法解决。这引出了其复杂性问题。我们证明：若区域$R$有限且单连通，则可通过一种称为"推进曲面算法"的算法求解该拼图，其复杂度为$O(|\partial R|^3)$，其中$\partial R$为区域边界的规模。当区域为整个无限三角网格时，我们证明可通过复杂度为$O(|X|^3)$的算法判定解的存在性，其中$X$为输入边集。为证明这些定理，我们重新审视了威廉·瑟斯顿关于区域$R$可铺砌性的研究成果。解法涉及卡里松铺砌、阶梯曲面与特定有向无环图割集之间的等价关系，该割集避免经过称为"不可破"的边集。这使我们能够推广瑟斯顿定理，通过重写为下降路径或吸收环来刻画可铺砌区域。瑟斯顿算法本质上是遵循迪杰斯特拉范式的距离计算算法。引入内部边集$X$会产生负权值，从而需要优先采用贝尔曼-福特策略。这些结论通过引入计算机科学的结构与算法，延伸了瑟斯顿的学术遗产。