MathDB

In an

n

n

grid, each cell is filled with an integer between

1

and

6

. The outmost cells all contain the number

1

, and any two cells that share a vertex has difference not equal to

3

. For any vertex

P

inside the grid (not including the boundary), there are

4

cells that have

P

has a vertex. If these four cells have exactly three distinct numbers

i

j

k

(two cells have the same number), and the two cells with the same number have a common side, we call

P

ijk

-type vertex. Let there be

A_{ijk}

vertices that are

ijk

-type. Prove that

A_{123}\equiv A_{246} \pmod 2

P4