How Many Uniquely Enumerated $4\times4$ Sudoku Grids Exist?

Not all of your $384$ possibilities actually work. For example one of your $384$ choices seems to be

$\begin{matrix} 1&2&|&3&? \\ 3&?&|&2&? \\ -&-&+&-&- \\ 2&3&|& ? & ? \\ ?&?&|& ? & ? \end{matrix}$

forcing

$\begin{matrix} 1&2&|&3&4 \\ 3&4&|&2&1 \\ -&-&+&-&- \\ 2&3&|& 1/4 & ? \\ 4&1&|& ? & 2/3 \end{matrix}$

but it is impossible to complete the grid.

Your argument about $8$ symmetries might fail if there is a pattern which is rotationally symmetric with a $180^\circ$ turn, which there is:

$\begin{matrix} 1&2&|&3&4 \\ 3&4&|&1&2 \\ -&-&+&-&- \\ 2&1&|& 4 & 3 \\ 4&3&|& 2 & 1 \end{matrix}$

Tags:

Combinatorics

Sudoku

Permutations

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