Princeton's campus culture is colorful, rich and varied, but among all the cultural exchanges and activities, there is no doubt that the afternoon tea discussion in Princeton's mathematics department is the one.

There are two types of afternoon tea discussions, one is a more formal professorial discussion that takes place every Wednesday afternoon in the West Lounge of the Department of Mathematics at Princeton University. The wives of the head of the department and other professors wore long gowns and white gloves. Serve everyone with heavy silver kettles and British bone china tea sets. In general, no students participated; The other is held at the rest of the time, in the opposite direction, in the East Lounge. It's a gathering between students, but it's usually attended by professors. Teachers and students, students chatting with each other, communicating freely, exchanging views on academic issues, discussing mathematical problems, and even exchanging some gossip in the mathematics world. During the party, students also play board games such as backgammon, military chess, chess, bridge, Critch cards, and a game named after John Nash.

During the afternoon tea discussion, the teachers and students of the Princeton Department of Mathematics can discuss cutting-edge mathematical problems together, so as to always maintain a keen and understanding of mathematics, so as to be able to achieve great achievements in mathematics, which is a major feature of Princeton.

Afternoon tea discussions are held almost every day, and although only the afternoon tea discussions between professors on Wednesdays are the most high-end discussions, they also attract most professors to participate in regular seminars, and often a student's casual idea or idea can inspire professors and open a new path for their research.

Professors at Princeton never treat students as ordinary students, but rather put them in a more equal peer that can discuss math problems, which is very tempting for students at Princeton. Almost all the students are looking forward to the Afternoon Tea Seminar, which is a platform for them to express themselves and keep track of the world's top math levels.

Regarding the afternoon tea discussion, Junxin has participated in it more than once, and naturally understands many of them very clearly, so he also has a great interest in the afternoon tea discussion. But what intrigues Junxin is the fascinating board games. This is also a special campus culture at Princeton.

Princeton's board games are naturally different from those in other places, and three people are required to participate at a time, one of whom acts as a referee and the other two as opponents. When playing a game, the opponents are in a blind match, that is, the two players play chess with their backs to each other, and only the referee knows the steps and situation of the two. The referee only reminds the captured piece and does not explain anything until one of them concedes. This not only tests the computing ability of both opponents, but also the memory skills of both opponents.

The board games at Princeton University's afternoon tea discussion were very popular, and not only the students actively participated, but also the professors actively participated in the board and card games at Princeton. And every participant hopes to be able to design a board game named after themselves to be recognized by everyone, the most famous of which is the legendary mathematician of Princeton University, Nobel laureate in economics John Nash's John Nash game, in addition to some games such as the game of life, etc., the whole board game is very interesting.

Since Princeton is relatively small, it is only natural that there are fewer graduate students than other Ivy League universities. Therefore, there is no university in the world that can compare with Princeton in undergraduate education, which is the most important reason why Junxin chose Princeton.

"Jun, aren't you going to play a game?" A white boy asked. He is about the same age as Junxin, and he is about seventeen or eighteen years old, but he is famous in the mathematics department of Princeton University, at least those professors can remember his name accurately, because his name is the same name as the first Fields Medal winner, the famous Finnish mathematician, the Als-Alfors who once took his Fields gold medal, although his mathematical talent is outstanding, but after all, he has not reached the height of the mathematician with the same name and surname.

It's also because of the name, plus he's also a roommate of Junxin, although the two are not in the same grade, but because of the same interests, they quickly became familiar with each other, and Junxin's first afternoon tea party was brought by Als Alfors.

"Not that interest for the time being!" Junxin stretched his waist and replied, and after being a referee for two games, he rejected Als's proposal, "I'm thinking about a question." ”

"Oh? What's the problem? Als asked with interest. Although board games are very popular in afternoon tea discussions, afternoon tea is a discussion based on academic discussions.

"It's not an academic problem!" Junxin waved his hand and said, "It's a question about chess and cards themselves, do you want to hear it?" ”

"Then tell me about it!"

"The study of geometry extends the concept of geometry to higher dimensions, and derives a series of rules and concepts of higher dimensions. So I wondered, can chess and cards also be extended to three-dimensional space, from the traditional plane to three-dimensional space? ”

"Huh? It's a good idea. Als immediately understood the meaning of Junxin's words, and asked excitedly, "Do you have any good ideas?" ”

"Take chess, for example." Jun Xin said, took a deck of chess, "Chess is a very typical two-dimensional chess game, 32 chess pieces move between 64 squares, we can use mathematical methods to estimate the total number of moves and the possibility of various moves." I'm thinking if I can transform it into a three-dimensional space chess game, set some rules, and even add some chess pieces when necessary, so as to design new ways to play and even new chess games, what do you think? ”

"You can give it a try!" Without hesitation, Als replied, "But how are you going to change it?" ”

"Ha, I haven't thought about it yet!" Junxin said with a smile, "But I already have a clue." ”

"Tell me about it!"

"We know that in the study of high-dimensional irregular geometry, mathematicians developed a new mathematical tool, topology. Nowadays, topology has become a major branch of mathematics. ”

"Topology is the study of the quantities in the high-dimensional geometry that do not change in the geometric transformation, can I imagine chess as a plane of geometry, and want to make them high-dimensional transformations, as long as I find the invariant quantities in it, then can I construct it into a three-dimensional space and make a three-dimensional chessboard and chess pieces?"

"I think," Ars muttered, touching his chin, "you can discuss this with Professor William Thurston." ”