Is it possible to write numbers in the cells of a $7\times7$ board in such a way that the sum of numbers in every $2\times2$ or $3\times3$ square is divisible by $1999$, but the sum of all numbers in the board is not divisible by $1999$?
Source: Ukraine 1999 Grade 8 P2
Tags: number theory, combinatorics
Is it possible to write numbers in the cells of a $7\times7$ board in such a way that the sum of numbers in every $2\times2$ or $3\times3$ square is divisible by $1999$, but the sum of all numbers in the board is not divisible by $1999$?
