\(10\)-by-\(10\) grid game?
You are a host of a game in which players are given a 10-by-10 grid, and are allowed to fill all of the squares on this grid with natural numbers from 1 to 10, as long as any numbers that share the same side or corner must be coprime.
Players win the game when they finished filling the grid with numbers, and there are no numbers appears more than \(a\) times in the grid. If any players win, you'll have to pay them 1000000 dollars.
The question is, what is the largest value of \(a\) you can define, such that there is no chance of winning to the players?