Passion for Puzzles has a newly posted puzzle that was found on beer coaster in Belgium. The puzzle is a 10X10 grid with filled circles and open circles in various squares. The goal is to fill in more circles so that there are the same amount of filled circles and open circles in each row and column. I’ve only looked at the puzzle briefly, but it’s very interesting. I’m curious to know if it has a single unique solution. Also, if it can be solved purely by logic or requires trial and error.

If there is a unique solution, then in that solution, every square must have a circle in it.

Indeed, given any solution in which some squares remain empty, we can add circles to some of the squares maintaining the equality of the number of filled circles and open circles remaining.

PROOF:

Each row or column which has an empty square in it, has at least 2 empty squares – since the grid is 10 by 10, and an even number of squares are filled in each row/column.

Let us choose a row r_1 that has an empty square, and then choose a column c_1 which meets r_1 at an empty square. Then choose a row r_2 that is not r_1 but meets c_1 at an empty square. Similarly, let c_2 be not c_1 but meet r_2 at an empty square. Continue.. Let r_n be a row that is not r_{n-1} but meets c_{n-1} at an empty square, and let c_n be a row that is not c_{n-1} but meets r_{n-1} at an empty square.

We can continue doing this as long as we wish, as each row/column with an empty square has at least two empty squares. Eventually we will repeat. Suppose that r_i=r_j is the first repeat in the rows with i