how do puzzle makers ensure that a sudoku board has one unique solution?
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Mathematics provides multiple ways to search for valid sudoku arrangements, and in some cases there are very fast methods — but even the slowest (brute-force guess-and-check) can be done so quickly by a computer that it would find singular solutions in seconds, or tell you if there is more than one from a given layout.
The difficulty with computers and creating singular sudoku starting positions is actually in finding starting grids that are easy vs intermediate vs challenging difficulty. Typically the software will have a number of rules for deduction or varying solve routines and depending on which was needed for solving or amount of look-ahead, determine the difficulty from that.
Some solvers (software) will do a partial solution up to a certain point where there is singularity then remove numbers from the grid one by one until solving it becomes more challenging.
Some people still write them by hand and just use an expert solver to check that it’s singular. There are some wild sudoku variants out there that probably started as an abstract exploration down some graph-theoretical journey.
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