1.3. Inna has two closed boxes - a square one and a round one. She was told that the round box contains 3 white and 10 black balls, while the square box contains 8 black balls. In one move, Inna can, without looking, take a ball from any box and either throw it away or move it to the other box. Inna wants to make the contents of the boxes the same. What is the minimum number of moves Inna can guarantee to achieve this?
Answer: 17

Solution. Suppose that in both boxes, there are both white and black balls left in the end. Then the last action Inna took was to draw a ball of a certain color from one of the boxes, and there were still both black and white balls left. However, she could not have done this with certainty, i.e., she might have kept drawing balls of the wrong color. Therefore, in the end, there will be balls of only one color left, and the other color needs to be removed.

If removing black balls, it would require a minimum of 18 actions, and if removing white balls - a minimum of 13 actions, since white balls can only be found at the bottom of the first box. At this point, there will be from 8 to 28 black balls left in the second box, and to separate them, it will require at least 4 more actions. Example: remove everything from the first box, and then transfer 4 balls from the second box to the first.