# Is this generalized Sudoku solvable?

Sudoku is played on a 9x9 grid where you place the digits 1 through 9 in individual grid cells.

The constraints are that every column must have all digits 1-9 and so must every row. On top of that, the 9x9 grid is broken up into a 3x3 grid of areas that each 3x3, which must also have all digits 1-9.

I'm wondering, if you add the requirement that ANY 3x3 section must contain all digits 1-9, is that solvable, or do the constraints make it impossible to arrange numbers to satisfy them?

Say you have a $1$ in the top left corner (row $1$, column $1$). Then given that it needs to occur in the box whose top left corner is $(2,1)$, and given that it cannot occur twice in the box whose top left corner is $(1,1)$, and finally given that it cannot occur in $(4,1)$ (for it cannot occur twice in column $1$), it will have to occur in either $(4,2)$ or $(4,3)$. Likewise, by considering row $1$ and the box whose top left corner is $(1,2)$ you need a $1$ in $(2,4)$ or $(3,4)$. Hence it is twice in the box whose top left corner is $(2,2)$.
• @AlanWolfe: you need to define what the requirement are for other sized grids. It seems obvious to me what the requirements are for a grid that is $n^2$ on a side, but $3 \times 3$ doesn't fit the pattern. For a $16 \times 16$ grid, I would expect it to be divided into $4 \times 4$ squares, each with the numbers $1$ through $16$. Dec 10, 2017 at 3:41