Tag Archives: basic

Thoughts on a Problem III

Posted on February 8, 2013 by limsup

I saw an interesting problem recently and can’t resist writing it up. The thought process for this problem was exceedingly unusual as you’ll see later. First, here’s the source: But here’s the full problem (rephrased a little) if you’d rather … Continue reading →

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Casual Introduction to Group Theory (1)

Posted on September 21, 2012 by limsup

Introduction Last year, I created a blog which was supposed to introduce concepts to abstract algebra in a systematic manner. Though I was reasonably happy with the end result, I got the sneaky feeling upon completion that the end product … Continue reading →

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Combinatorial Game Theory V

Posted on August 10, 2012 by limsup

Lesson 5 We did mention in the first lesson that CGT covers games without draws. Here, we’ll break this rule and look at loopy games, i.e. games with possible draws. [ To be specific, loopy games are those where it’s … Continue reading →

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Combinatorial Game Theory Quiz 1

Posted on August 8, 2012 by limsup

This quiz lasts 70 minutes and covers materials from lessons 1-4. For A-C, determine whether the following Nim games are first or second-player wins. There is no need to find the winning move. (10 points) (10, 15, 17, 19) (7, … Continue reading →

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Combinatorial Game Theory IV

Posted on August 6, 2012 by limsup

Lesson 4 In this lesson, we will work on a large class of games, known as take-and-break games. First consider a simple example. Kayles Kayles is an example of a take-and-break game: Start with a few heaps of contiguous bottles, … Continue reading →

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Combinatorial Game Theory III

Posted on August 5, 2012 by limsup

Lesson 3 We’ve learnt Nim and we’ve learnt the Square Game. Now, let’s combine them and consider the following game, which we shall name Nim Square. Start with r heaps of stones, of sizes . Play alternates between two players: at … Continue reading →

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Combinatorial Game Theory II

Posted on August 4, 2012 by limsup

Lesson 2 In this lesson, we will focus on a special type of game called Nim. Although it’s only one out of infinitely many possible games, understanding it in depth will be very beneficial in analysing a much larger class … Continue reading →

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Combinatorial Game Theory I

Posted on August 2, 2012 by limsup

[ Prerequisites required: none for now. ] In the middle of 2000, I was waiting to go to graduate school and had a bit of free time on my hands. So I decided to prepare a website which teaches combinatorial … Continue reading →

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Power Series and Generating Functions (I): Basics

Posted on February 5, 2012 by limsup

[ Background required: basic combinatorics, including combinations and permutations. Thus, you should know the formulae and and what they mean. Also, some examples / problems may require calculus. ] Note: this post is still highly relevant to competition-mathematics. 🙂 To … Continue reading →

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Symmetric Polynomials (III)

Posted on December 20, 2011 by limsup

Now we generalise this to n variables: . It’s clear what the corresponding building blocks of symmetric polynomials would be: ; ; ; … . We call these ei‘s the elementary symmetric polynomials in the xi‘s. Note that each ei is the coefficient of Ti in the … Continue reading →

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