The mathematics of Septoku

TL;DR

This paper analyzes Septoku, a hexagonal Sudoku variant, revealing the limited number of valid boards, the minimal clues for uniqueness, and generalizing to new shapes, including a star-shaped puzzle connected to a geometric problem.

Contribution

It characterizes all valid Septoku boards, determines the minimal clues for unique solutions, and introduces new puzzles on novel shapes linked to geometric problems.

Findings

Only six valid Septoku boards exist up to symmetries.

Six clues are necessary for a unique solution.

A new star-shaped puzzle with 73 cells has a unique solution.

Abstract

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.