The last ball

1. The final color does not depend on the specific sequence of extractions.
2. Notice what simple feature of the white balls does or does not change in each rule.
3. If this trait is preserved, the last ball is already very conditioned.

Answer: the final color depends only on the initial parity of white balls:

• white even initials -> last black ball;
• odd initial white -> last white ball. It does not depend on the extraction order. Proof by invariant: White's parity does not change:
• two white ones -> you remove 2 white ones and add black one: change -2;
• two black -> change 0 to white;
• one white and one black -> you remove one white and add one white: change 0. White parity is always preserved. In the end there is only one ball left: if it is white, there is 1 white (odd); if it is black, 0 white (even).