10.2. There are 2004 small boxes on the table, each containing 1 ball. It is known that some of the balls are white, and there are an even number of white balls. You are allowed to point to any 2 boxes and ask: "Do they contain at least 1 white ball?" How many times do you need to ask, at a minimum, to determine a box that contains a white ball?
10.2. 2003.

Number the boxes (and the balls inside them) from 1 to 2004, and refer to the problems by the numbers of the box pairs, and call all non-white balls black balls.

First, we prove that a white ball can be found with 2003 questions. Ask the questions $(1,2),(1,3), \cdots,(1,2004)$ in sequence. If all the answers are affirmative, then ball 1 is white (in fact, if ball 1 were black, then there would be at least one more black ball that forms a black pair with ball 1). If at least one of the answers is negative, then ball 1 is black. In this case, the other ball in the pairs that include ball 1 and received an affirmative answer must be white. Thus, all the white balls are identified.

Second, we prove that fewer than 2003 questions cannot suffice. Assume there exists a strategy that ensures finding a white ball with fewer questions. We can answer all the questions affirmatively. At most, after 2002 questions, one box (let's say the first box) must be identified as containing a white ball. At this point, there must be a question of the form $(1, n)$ that has not been asked, for example, question $(1,2)$, otherwise, 2003 questions would be needed. If the first and second boxes contain black balls, and all other boxes contain white balls, then all our answers are correct. However, the ball identified by the player is black, leading to a contradiction.