1.1. Polina has two closed boxes - a square one and a round one. She was told that the round one contains 4 white and 6 black balls, while the square one contains 10 black balls. In one move, Polina can take a ball from any box without looking and either throw it away or move it to the other box. Polina wants to make the contents of the boxes the same. What is the minimum number of moves Polina can guarantee to achieve this?

![](https://cdn.mathpix.com/cropped/2024_05_06_384ab8f17223d7967932g-01.jpg?height=482&width=446&top_left_y=1164&top_left_x=879)
Answer: 15

Solution. Suppose that in the end, both boxes still contain both white and black balls. Then the last action Polina 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. But she could not have done this with certainty, meaning she could 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 must be removed. If we draw a ball from one of the boxes and put it back, the action has no effect, so with each draw, it is necessary to either move or remove the ball.

If we remove the black balls, it will take a minimum of 16 actions, and if we remove the white balls, it will take a minimum of 10 actions, since the white balls can only be found at the bottom of the first box. In this case, there will be between 10 and 16 black balls left in the second box, and to separate them, it will take at least 5 more actions. Example: remove everything from the first box, and then move 5 balls from the second box to the first.