For mathematics, some people will think that it is good to let the people they like to study, and it has nothing to do with themselves, and some people will think that it is very beautiful and rational, and if they have time, they also want to take it as a hobby.
However, the mathematics presented in this book is not the same as in either case. This book introduces the kind of mathematics that may not be beautiful, but anyone can use it in life, and the mathematical ideas behind it. In fact, it is more appropriate to call it "engineering mathematics" rather than "mathematics", that is, "using mathematics as a tool to solve problems in engineering".
In this book, I will introduce you to some useful mathematical and mathematical ideas in daily life. "Useful" is the focus of this book, and the topics listed in the book are all from the perspective of solving practical problems. There are some topics that, although mathematically interesting, only satisfy people's curiosity and do not solve practical problems, and I try not to touch on them.
On the contrary, as long as I can solve a real problem, even if someone says, "Is this math?", I will include these topics. Of course, the result of this is that the topics are varied and divergent, and it does not form an organized system, but it will undoubtedly make everyone feel the impact of "it turns out that mathematics is really useful".
How to get the best result" – this is a question that must be considered in any competition.
Of course, if you're far superior to your opponent, there's no need to think about how your opponent will move. However, the two sides are usually evenly matched, and in such cases, it is necessary to change the strategy according to the actions of the opponent. Whether it's in the mall or on the field, this is universal.
In baseball, for example, when the batsman has already had two good balls, the next step is to choose between the fastest reaction to the straight ball or the patient waiting for the changing ballEven if the batsman is very good at playing the variant ball, if a straight ball suddenly comes, it is likely to be a strikeout. On the other hand, if you choose to deal with a straight ball, if you come to a change ball that you are good at, it is easy to miss the opportunity by hitting ......a throw-in
In this way, the strategy of maximizing losses by judging the actions of the opponent to increase one's own gains is called game theory. One example of game theory that we are most familiar with is "rock, paper, scissors".
Rock, paper, scissors" is also "a way to increase your chances of winning."
It stands to reason that "rock, paper, scissors" should be a fair game, and there is no "sure-fire way". However, there is still some room for effort when it comes to "improving the win rate". It's not enough to play only once, if you play with the same person a few more times, you can summarize your opponent's habits and use this to improve your own winning rate.
Let's say you happen to watch Little A play with another person. Here, we use "S" for rock, "J" for scissors, "B" for paper, and then "win", "loss" and "draw" respectively to indicate the outcome of the game. For example, a win out of a stone is counted as "s win", a loss out of scissors is recorded as a "j loss", and a draw out of a cloth is recorded as "b draw".
By observing the course of Little A's 9 rounds of play, it was found that the results are as follows:
That is to say, Little A won in the 1st round out of the stone, out of the scissors draw in the 2nd round, lost in the 3rd round of the cloth, and won the 4th round of the cloth but won the ......From this data, we need to analyze the habits of Little A and use this to improve our winning rate. I wonder if you have any good ideas
Analyze your opponent's habits.
The easiest way is to count the number of occurrences of rock, scissors, and paper, and it turns out that the stone appears 3 times, scissors appear 2 times, and paper appears 4 times, so it can be concluded that Little A likes to produce cloth the most and dislikes scissors the least. However, we only looked at 9 rounds of the game, and if you want to use this method, it's best to draw conclusions from a longer-term observation. Moreover, if you find that Little A likes to make cloth the most, you will only take out scissors, and Little A will soon notice it.
If we want to know more about Little A's habits, we can analyze "what will happen next time after Little A is out once". We can follow Table 1-1.
Make a 3-row, 3-column **, label each column with s, j, b, to indicate "what happened this time", and then mark s, j, and b on each row to indicate "what happened next time".
Table 1-1 analyzes "what happens next after one small out".
In A's 9 shots, we only focus on "what happened this time" and "what happened next time". Little A first played S and then J, so we were.
Add 1 vote to the grid where column 1 (indicating this outing) and row 2 (indicating the next outing) intersect. Little A is out of J and then out of B for the second time, so we're in.
Add 1 vote to the grid where column 2 and row 3 intersect. By analogy, we can count it based on the data of Xiao A's 9 shots.
8 votes in the result.
By looking at this result, we can see that j comes out 2 times after s and b after b there are 2 times. In addition, there is no case of 1 occurrence of s after s. This table looks useful, and if you can look at it for a few more rounds and accumulate more data, you can further increase its credibility. Compared to the previous method, it is difficult for Little A to detect that his opponent has analyzed his shooting habits, so it is likely that he will continue to repeat the shooting strategy he is used to.
What the opponent likes to play and what he doesn't like to play.
We can also think further, on the basis of "what will happen this time, what will happen next time", and take into account the results of victory and defeat, it will be easier to judge how we should make a move.
To do this, we can make a ** with 3 rows and 9 columns as shown in Table 1-2. The columns in represent the grouping of "This Shot and Its Outcome and Results", and the rows indicate the next shot. Will just now.
This table is the result in Table 1-2. Similarly, if more data can be collected, it will be possible to analyze Little A's shooting habits in more detail.
Table 1-2 adds the results of Wins, Losses, and Draws to the table.
In this way, by statistically stating the habits of your opponents and analyzing what your opponents like and don't like, you can come up with better coping strategies. However, even if the statistics show that Little A likes to make the cloth the most, you can't just take out the scissors, otherwise Little A will notice it and change his strategy. Therefore, a better strategy should be to pretend that there is no pattern in the shot, but secretly make a few more scissors, so that you should be able to win a few more times.
The use of "information from the last shot" is "conditional probability".
Under the condition that "the opponent made a stone last time", the probability that "what will happen next time" is called conditional probability. In the case where the previous state (e.g., the opponent last shot out of rock) is determined, the "conditional probability" of the various states that may occur next (e.g., the opponent out of rock, scissors, or paper) is determined, and such a process is called the Markov process.
The method we have presented here is to assume that the opponent's actions are Markov processes and make decisions based on observations of conditional probability. However, in order for this method to produce better results, it is necessary to accumulate a large amount of observational data, which means that it is very important to make efforts to collect data.
An analysis of the rock, scissors, and paper sections of "Miss Conch".
It's amazing to hear of people who have actually practiced such a method. There is an organization called the Miss Conch Rock, Scissors and Paper Institute. For the "rock, paper, scissors" section at the end of the cartoon Miss Conch, this institute collected data from the last 25 years, achieving a score of 78 in 20155% win rate.
The data analysis method they use is not limited to "the information of the last shot", but based on the past two shots, the current week's shots, and find some patterns, such as "it is rare to have the same shot 3 times in a row", "most of the scissors are out of the first episode when the new season of the show starts", etc. If Miss Conch's shot is completely random, then it will be difficult to carry out**, but in fact, what is decided by the staff behind the scenes, so there must be some "habits". It seems that there are many doorways in "rock, scissors, and paper"!
Recommended Reading: 
What is the Purpose of Mathematics: How to Use Mathematics to Solve Practical Problems
Author: [Japanese] Atsuyoshi Sugihara.
Translator: Zhou Ziheng.
This book is here to answer a puzzle: what is the use of mathematics?
Computation, Geometry, Probability, Graph Theory, Game Theory, Vectors, Modeling, ......How to integrate mathematical thinking into life and learning?
From workplace strategies to daily games, from mental exploration to fitness exercises, from writing skills to tips for practicing calligraphy, fun and beneficial math is everywhere.
As long as you have the level of mathematics knowledge in primary and secondary schools, you can experience the fun.