Task 2. Pete liked the puzzle, he decided to glue it together and hang it on the wall. In one minute, he glued together two pieces (initial or previously glued). As a result, the entire puzzle was joined into one complete picture in 2 hours. How long would it have taken to assemble the picture if Pete had glued together three pieces per minute instead of two?

[5 points] (A. V. Shapovalov)
Answer: In one hour.

Solution 1. Each gluing reduces the number of pieces on the table by 1. Since after 120 gluings one piece (the complete puzzle) was obtained, there were 121 pieces at the beginning. Now, if three pieces are glued together per minute (i.e., the number of pieces is reduced by 2), one piece will remain after 60 minutes.

Solution 2. In one minute, Petya made one gluing and spent 2 hours assembling the puzzle. If he glued three pieces together, he would make two gluings per minute, i.e., work twice as fast and thus spend 1 hour.

Comment. The time spent gluing the entire puzzle does not depend on the order in which Petya took the pieces. For example, he could add one piece at a time to the already glued part. Or he could glue the pieces in pairs or triplets and then glue them together. In the first case, it would take him 2 hours, and in the second case, 1 hour. In the first solution, this fact is proven by counting the total number of pieces. In the second solution, it is taken as obvious. Therefore, strictly speaking, the second solution is incomplete: it provides the correct answer, but does not prove its uniqueness.