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In combinatorics, the Pigeonhole Principle is a fundamental tool used to prove the existence of certain properties within a finite set of objects. Essentially, this principle states that if you have more 'pigeons' than 'pigeonholes' to place them in, then at least one pigeonhole must contain more than one pigeon. In our exercise, the 21 girls and 21 boys represent the 'pigeons', and the questions they solve represent the 'pigeonholes'. Each student solves up to six questions, which adds up to a total of 252 question-solving instances (126 for girls and 126 for boys). Since each pair of a boy and a girl shares at least one question, we use combinatorial analysis to determine that at least one of these questions must be common across several participants. This is the crux of how combinatorics and the Pigeonhole Principle help solve the problem.

In set theory, an overlap between sets indicates elements shared by multiple sets. For this exercise, consider each question as a set where each set contains the students who solved that question. Since every boy-girl pair solves at least one common question, we analyze the overlap among these sets. Each of the 21 girls and 21 boys can solve at most six questions, giving us a total of 252 distinct question-solving instances. Due to the conditions of the problem, these instances must be distributed in such a way that some questions are solved by multiple boys and girls. Technically, this overlap creates intersections between the sets corresponding to each question. By calculating the intersections, we prove that there must be a question solved by at least three girls and three boys, ensuring the constraints are met. The result of this overlap becomes evident through the application of combinatorial principles.

Mathematical proofs involve rigorously demonstrating that a given statement logically follows from assumed axioms or previously established theorems. Here, to prove that a question is solved by at least three girls and three boys, we use the Pigeonhole Principle methodically. First, we established that there are 252 question-solving instances among the 42 students. Because each student can solve up to six questions, distributing these instances equally would mean each question is associated with multiple students. Now, applying the Pigeonhole Principle, if we distribute these 252 instances over the questions, there must be some questions solved by more than a minimal number of participants. By calculating and considering the overlaps, we show that at least one question is indeed solved by a minimum of three girls and three boys. The structured approach taken in the proof ensures all logical steps are clearly demonstrated and easily understood.