To make the cup face down, you must "flip" it an odd number of times. To make all nine cups face down, you must "flip" the sum of nine odd numbers. In other words, the total number of "flips" is odd.

Introduction to Olympic mathematics

Olympic Mathematics is the abbreviation of Olympic Mathematics.

1934 and 1935, the former Soviet Union began to hold middle school mathematics competitions in Leningrad and Moscow, and named them Mathematical Olympics. 1959 Romanian mathematical physics society invited middle school students from eastern European countries to participate in the first international mathematical olympiad held in Bucharest. Since then, it has been held once a year and has been held for 50 times.

The scope of Olympic Mathematics is beyond the level of compulsory education in all countries. Some topics are much more difficult than the college entrance examination, and some topics are even difficult for mathematicians. Through such a high-level competition, we can find mathematical talents as early as possible, and then train them to make them stand out.