Ask HN: Can you prove that my puzzle is always solvable with unlimited moves?

Gameplay video: https://www.youtube.com/watch?v=ACON8hRxbdU

Tutorial video: https://www.youtube.com/watch?v=eV-lnxyLwYk

Tile Wipeout Game Rules

The game is played on a 6×6 grid with 6 colors. Each color appears 6 times: 1 circle and 5 squares.

Circles are fixed in place. They cannot be moved or removed. Every row and every column contains exactly one circle.

Your goal is to leave the grid as empty as possible within the given number of moves.

Movement

Swipe any row or column to rotate it by one position.

Only the object that passes the row's or column's circle is affected. All other objects simply move with the rotation.

Circle interactions

When an object passes a circle:

An unshielded square matching the circle's color is removed, leaving an empty space. A shielded square matching the circle's color is not removed. An unshielded square of a different color gains a shield. A shielded square of a different color loses its shield. An empty space becomes a new square of the circle's color.

Shielded squares are shown with a dark interior.

Within each color, larger squares are closer to their matching circle.

Ending the game

You may end the game at any time. Removing every square may not be possible.

Your score is:

(% empty × 1000) + moves remaining

where % empty is the percentage of non-circle cells that are empty.