Football is known as the first sport in the world today, and the world-famous quadrennial World Cup is currently in full swing in South Korea and Japan. Some seed teams were eliminated in the group stage, "dark horses" emerged constantly, and the pattern of world football gradually changed. Usually, when people watch a football match, they mainly appreciate the players' fierce confrontation, skillful footwork, excellent cooperation and accurate shooting. This paper attempts to discuss several problems in the World Cup from the perspective of mathematics.

Material 1: According to the rules of the World Cup group match, 4 teams in each group play a single round robin match, with 3 matches for each team and 6 matches for each group. In each game, the winning team scored 3 points, the losing team scored 0 points, and the teams tied 1 point. After the group stage, the two teams with the highest points advanced. If the scores are the same, the team with more goal difference wins.

Question: How many points does the team definitely qualify or basically qualify?

(Figure 1)

Material 2: In the past, football was mostly a polyhedron made of black and white leather, in which the black leather was a regular pentagon and the white leather was a regular hexagon. The surface has the following characteristics: (1) Black leather is surrounded by white leather; (2) Every two adjacent regular polygons have exactly one common edge; (3) Each vertex is the common edge of three adjacent skins, and it is black and white (as shown in figure 1).

With the development of science and technology, the improvement of football sports level and the change of people's aesthetic standards, the selection of football leather and the production methods have been greatly improved. For example, the football "Fly to fireball" in this World Cup adopts the latest technology, and the surface of the ball is designed with honeycomb foam.

The weight of football is reduced, the flying speed is faster, and the coating on the surface will appear faint gold under the irradiation of light, making football a real work of art, but at the same time, we find that the structural characteristics of its regular pentagon and regular hexagon are the same.

Question: (1) The number of regular pentagons and regular hexagons; ⑵ The relationship between sphere and regular polyhedron.

Material 3: Before the start of the World Cup in Korea and Japan, two promoters of World Cup mascots, Party A and Party B, sold mascots in the same place at the same price every time (the price of mascots may vary with time). Two months, one month and one day before the start of the competition, they publicized it three times each. A sells 500 mascots at a time, and B earns 500 dollars at a time. Now it is stipulated that whoever sells more money per mascot will have a good sales method.

Question: Who has the better sales method, A or B?

Material 4: There are 32 teams participating in the Korea-Japan World Cup. Divide into 8 groups for single round robin, and select the top 65,438+06 teams. This 16 team was eliminated according to certain procedures, and finally the champion, runner-up and third and fourth place were selected.

Question: (1) The total number of games to be arranged; ⑵ A local TV station held a quiz draw. The method is as follows: after 16 decides the top 8, but before the top 4 decides, let fans guess the top 4 and the last one, two, three and four. How likely are students to answer all the questions correctly in the test?