## COMP418 Combinatorics课程简介

Applied mathematics is the mathematical study of general scientific concepts, principles, and phenomena that, because of their widespread occurrence and application, relate or unify various disciplines. The core of the program at MIT concerns the following principles and their mathematical formulations: propagation, equilibrium, stability, optimization, computation, statistics, and random processes.

Sophomores interested in applied mathematics typically survey the field by enrolling in 18.200 and 18.300 Principles of Applied Mathematics. Subject 18.200 is devoted to the discrete aspects of the study and may be taken concurrently with 18.03. It carries CI-M credit in mathematics. Subject 18.300, given only in the second term, is devoted to continuous aspects and makes considerable use of differential equations.

The subjects in Group I of the program correspond roughly to those areas of applied mathematics that make heavy use of discrete mathematics, while Group II emphasizes those subjects that deal mainly with continuous processes. Some subjects, such as probability or numerical analysis, have both discrete and continuous aspects.

Students planning to go on to graduate work in applied mathematics should also take some basic subjects in analysis and algebra.

## PREREQUISITES

Combinatorics is a branch of mathematics with broad areas of application. There are important uses of combinatorics in computer science, operations research, probability, and statistics. Theoretical thermodynamics uses combinatorial theory to describe ideas such as entropy. The combinatorial analysis is a cornerstone of the study of error-correcting codes; these codes are used to transmit information from deep space or to protect the quality of music on compact discs. Our course will mainly focus on describing and/or counting complicated sets. Often questions that begin “How many ways can you…?” or “How many steps does it take to…?” are answered using combinatorial analysis. Such questions on the surface may appear rather uninteresting, but one can quickly get to questions that are quite engaging. What gambler wouldn’t want to understand the odds at winning a poker hand?

## COMP418 Combinatorics HELP（EXAM HELP， ONLINE TUTOR）

This game begins with 1000 counters. A turn consists of removing any power of 2 number of counters. Who wins and how do they win.

This is a classic example of the game called “The Game of Nim”. In this version of the game, the player who takes the last counter wins.

To see why, we can consider the following strategy:

• Suppose the number of counters remaining is a power of 2 (i.e., 1, 2, 4, 8, 16, 32, 64, 128, 256, or 512). In this case, the player whose turn it is can take all the remaining counters and win.
• Suppose the number of counters remaining is not a power of 2. In this case, the player whose turn it is can always remove enough counters to make the number of counters remaining a power of 2. For example, if there are 12 counters remaining, the player can remove 4 counters to leave 8 counters remaining. Then, the other player must remove some number of counters, leaving a non-power-of-2 number of counters remaining, and the first player can use the strategy above to win.

Therefore, the player who goes first can always win by following the above strategy. They can simply ensure that the number of counters remaining after their turn is always a power of 2, and then take all the remaining counters on their final turn to win.

This game is played with one knight on a chess board. The first player places the knight on the board. The second player then moves the knight to a new square; that is, to one which has not previously been occupied. Players alternate, each time moving to a new square. The first player who cannot move loses. Who wins with best play?

In this game, the second player can always win with best play, no matter how the first player places the knight on the board.

To see why, we can consider the following strategy for the second player:

1. After the first player places the knight on the board, the second player mirrors the first player’s move about the center of the board. That is, if the first player places the knight on square (i, j), the second player places the knight on square (9-i, 9-j).
2. On each subsequent turn, the second player mirrors the first player’s move about the center of the board. That is, if the first player moves the knight to square (i, j), the second player moves the knight to square (9-i, 9-j).

It’s easy to see that this strategy ensures that the second player never moves the knight to a square that has already been occupied by either player. Furthermore, if the first player cannot make a legal move, it means that all squares on the board have been occupied, and the second player has won.

Therefore, with best play, the second player will always win this game.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供luc.edu COMP418 Combinatorics组合学课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## COMP418 Combinatorics课程简介

Applied mathematics is the mathematical study of general scientific concepts, principles, and phenomena that, because of their widespread occurrence and application, relate or unify various disciplines. The core of the program at MIT concerns the following principles and their mathematical formulations: propagation, equilibrium, stability, optimization, computation, statistics, and random processes.

Sophomores interested in applied mathematics typically survey the field by enrolling in 18.200 and 18.300 Principles of Applied Mathematics. Subject 18.200 is devoted to the discrete aspects of the study and may be taken concurrently with 18.03. It carries CI-M credit in mathematics. Subject 18.300, given only in the second term, is devoted to continuous aspects and makes considerable use of differential equations.

The subjects in Group I of the program correspond roughly to those areas of applied mathematics that make heavy use of discrete mathematics, while Group II emphasizes those subjects that deal mainly with continuous processes. Some subjects, such as probability or numerical analysis, have both discrete and continuous aspects.

Students planning to go on to graduate work in applied mathematics should also take some basic subjects in analysis and algebra.

## PREREQUISITES

Combinatorics is a branch of mathematics with broad areas of application. There are important uses of combinatorics in computer science, operations research, probability, and statistics. Theoretical thermodynamics uses combinatorial theory to describe ideas such as entropy. The combinatorial analysis is a cornerstone of the study of error-correcting codes; these codes are used to transmit information from deep space or to protect the quality of music on compact discs. Our course will mainly focus on describing and/or counting complicated sets. Often questions that begin “How many ways can you…?” or “How many steps does it take to…?” are answered using combinatorial analysis. Such questions on the surface may appear rather uninteresting, but one can quickly get to questions that are quite engaging. What gambler wouldn’t want to understand the odds at winning a poker hand?

## COMP418 Combinatorics HELP（EXAM HELP， ONLINE TUTOR）

1. This game begins with the number 0 . Each turn consists of adding to the current number a whole number from 1 to 9 . The player who reaches 100 wins.

This game is commonly known as “Addition Game” or “The 100 Game”. Here are the basic rules:

1. The game begins with the number 0.
2. Each player takes turns adding a whole number from 1 to 9 to the current number.
3. The resulting number becomes the new current number.
4. The player who reaches 100 exactly wins the game.
5. If a player goes over 100, they lose the game.

For example, if the current number is 67 and it is the player’s turn, they can add any number from 1 to 9 to get a new current number. If they choose to add 6, the new current number becomes 73. The game continues until a player reaches exactly 100 or until someone goes over 100 and loses.

It is important to note that each player must add a different number to the current number on their turn. Players cannot repeat the same number that has already been used in the game.

This game begins with the number 1. A turn consists of multiplying the current number by a number between 2 and 9 . The first player who reaches a number greater than 1000 wins.

This game is commonly known as “Multiplication Game”. Here are the basic rules:

1. The game begins with the number 1.
2. Each player takes turns multiplying the current number by a whole number from 2 to 9.
3. The resulting number becomes the new current number.
4. The player who first reaches a number greater than 1000 wins the game.

For example, if the current number is 27 and it is the player’s turn, they can multiply by any number from 2 to 9 to get a new current number. If they choose to multiply by 3, the new current number becomes 81. The game continues until a player reaches a number greater than 1000 and wins.

It is important to note that each player must multiply the current number by a different number on their turn. Players cannot repeat the same number that has already been used in the game.

Also, players must be careful not to overshoot the winning number. For instance, if a player’s turn would result in a number greater than 1000, they lose the game.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供luc.edu COMP418 Combinatorics组合学课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。