The rules of Masyu:
Make a single loop with lines passing through the centers of cells, horizontally or vertically. The loop never crosses itself, branches off, or goes through the same cell twice.
Lines must pass through all cells with black and white circles.
Lines passing through white circles must pass straight through its cell, and make a right-angled turn on at least one side of the white circle cell (left or right).
Lines passing through black circles must make a right-angled turn in its cell, then it must go straight through the next cell (till the middle of the second cell) on both sides.
A valid Masyu puzzle is a rectangular grid with some squares containing circles such that a unique loop can be drawn according to the above rules.
What is the highest possible number of circles that can be placed on a chessboard to create a valid Masyu puzzle?
What is the highest possible proportion of squares that contain circles in a valid Masyu puzzle on any large square grid?