For a game of Sukoshu—which is a simpler version of Sudoku, consisting of one 4*4 board—the axiomatic expression of the rules is as follows:

1) ∀x.∀y.∀z.∀w.(cell(x,y,w) ∧ cell(x,z,w) ⇒ same(y,z))
This sentence expresses the constraint whereby two cells in the same row cannot contain the same value.

2) ∀x.∀y.∀z.∀w.(cell(x,z,w) ∧ cell(y,z,w) ⇒ same(x,y))
This sentence expresses the constraint whereby two cells in that same column cannot contain the same value.

And, 3) ∀x.∀y.∃w.cell(x,y,w)
This sentence expresses the fact that every cell must contain at least one value.

Now, a game of Sudoku consists of a 9*9 board divided into nine 3*3 subboards. We want to axiomatically express the rules whereby, when filling empty cells, no numeral must be repeated in any row or column or 3*3 subboard.
How must we proceed? Do we preserve the sentences 1, 2, and 3 from the axiomatisation of the rules of Sukoshu and merely come up with a new sentence for the rule whereby no number is to be repeated in any 3*3 subboard?