Septoku is a Sudoku variant invented by Bruce Oberg, played on a hexagonal grid of 37 cells. We show that up to rotations, reflections, and symbol permutations, there are only six valid Septoku boards. In order to have a unique solution, we show that the minimum number of given values is six. We generalize the puzzle to other board shapes, and devise a puzzle on a star-shaped board with 73 cells with six givens which has a unique solution. We show how this puzzle relates to the unsolved Hadwiger-Nelson problem in combinatorial geometry.

中文翻译：

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Septoku的数学

Septoku 是由 Bruce Oberg 发明的数独变体，在 37 个单元格的六边形网格上播放。我们表明，直到旋转、反射和符号排列，只有六个有效的 Septoku 板。为了有一个唯一的解决方案，我们证明给定值的最小数量是 6。我们将拼图推广到其他棋盘形状，并在具有 73 个单元和六个给定的星形棋盘上设计了一个具有独特解决方案的拼图。我们展示了这个谜题如何与组合几何中未解决的 Hadwiger-Nelson 问题相关。