In our class club afternoon, 31 of us participated, and after the program, we danced. Ági danced with 7 boys, Anikó with 8, Zsuzsa with 9; the girls can be listed further in such a way that each danced with 1 more boy than the one mentioned before; finally, Márta only did not dance with 3 boys. How many boys were there at the club afternoon?
Each girl danced with six more boys than her position in the list of girls. Márta is as many in this list as there are girls in total, and she danced with six more boys than this, and did not dance with three boys - so there are nine more boys than girls. If, at a moment, every girl were dancing, then there would be nine boys without a partner, 22 would be dancing, 11 pairs. Thus, there are 11 girls and 20 boys participating.

Tamás Endre (Budapest, I. István g. I. o. t.)

Note. We quote from two papers: "Let's assume there was a situation where Ági was dancing with the 7th boy, Anikó with the 8th, ... The last 3 boys did not get a girl, but the first 6 did not either..." - "Let's calculate the total number of participants minus 9 boys, or the 3 who did not dance with Márta and the first 6..."

The author of the first quote showed some caution with the "let's assume" introduction. Such assumptions, complementing the relationships stated in the text, can indeed facilitate the solution of many problems. However, such a solution is only complete if we show that our additional assumption does not contradict the original relationships.

Both the author of the first quote and numerous other contestants implicitly assumed that the boys were also somehow put in order, and, for example, all of Ági's dance partners danced with Anikó, and one more, similarly, all of Anikó's dance partners danced with Zsuzsa, and one more. This assumption is not impossible, but quite unlikely; conversely, it is also not impossible that Ági and Anikó had no common dance partners.

A similar assumption is the following: "those who did not dance with Márta could not have danced with any girl." Many papers, although their answer was correct, were based on such an assumption that is not exactly wrong, but uncertain. Despite the ease of the problem, 8% of the papers were incorrect.