Problem description:

In 2003, the last question of Shandong junior high school competition was about 12 people, and 13 flowers were randomly allocated to 12 people. If you have two or more flowers in your hand, you should give yourself one left and one right. Anyway, it will prove that at least seven people have flowers in their hands Please write down the analysis process. Thank you!

Analysis:

I found the answers to all the questions for you.

There are 12 students in a circle, and some have flowers in their hands. The total number of flowers is 13. They play a game of sharing flowers, and divide the flowers according to the following rules every time: one of the students who has at least two flowers in his hand takes out two flowers for two adjacent students, one for each. Tests show that at least seven students will hold flowers in their hands in the continuous flower sharing game.

Let's assume that there are less than seven students holding flowers at the beginning. We mark this 12 student counterclockwise with a, A2, a, …, A2 respectively.

(1) In the flower sharing game, once one of two adjacent students holds flowers, at least one of them will always hold flowers after each flower sharing. In fact, if the students who divide flowers are not one of the two students, the flowers in their hands will only increase, not decrease. If one of them is a flower distributor,

(2) It is impossible for any student to have no flowers in his hand all the time, which can be proved by disproof. Let's assume that A 1 has never had flowers in his hand, which means that A2 has never been a florist, and the flowers in his hand can only be increased, but not decreased. Because there are only 13 flowers in total, A2 will no longer accept flowers after a limited number of flowers are distributed. This means that A2 will no longer accept flowers after a limited number of flowers are distributed. After the flower limit, A4 will no longer be a flower dealer. By analogy, after the flowers are limited, none of all 12 students will be a flower distributor, and the activity will be terminated. This is inconsistent with the fact that 13 flowers are distributed among 12 students, and students who may be scored can always be found under any circumstances.

According to (1) and (2), at least one of any two adjacent students can have flowers after several times of flower matching, so at least six students have flowers. If only six students have flowers, students with flowers cannot be adjacent, otherwise two students without flowers are adjacent. So as long as the flowers are distributed again, at least one student with flowers will be added.