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Tag Archives: basic
Thoughts on a Problem III
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
Casual Introduction to Group Theory (1)
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
Posted in Notes
Tagged 15 puzzle, abstract algebra, basic, casual, conjugate, group theory, order, permutations, rubik's cube, symmetries
2 Comments
Combinatorial Game Theory V
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
Posted in Notes
Tagged basic, combinatorial game theory, computer science, impartial games, loopy games, nim values, programming
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Combinatorial Game Theory Quiz 1
This quiz lasts 70 minutes and covers materials from lessons 14. For AC, determine whether the following Nim games are first or secondplayer wins. There is no need to find the winning move. (10 points) (10, 15, 17, 19) (7, … Continue reading
Posted in Homework
Tagged basic, combinatorial game theory, computer science, impartial games, nim, nim values
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Combinatorial Game Theory IV
Lesson 4 In this lesson, we will work on a large class of games, known as takeandbreak games. First consider a simple example. Kayles Kayles is an example of a takeandbreak game: Start with a few heaps of contiguous bottles, … Continue reading
Posted in Notes
Tagged basic, combinatorial game theory, computer science, impartial games, nim values
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Combinatorial Game Theory III
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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Tagged basic, combinatorial game theory, computer science, impartial games, nim values, programming
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Combinatorial Game Theory II
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
Posted in Notes
Tagged basic, combinatorial game theory, computer science, impartial games, nim, programming
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Combinatorial Game Theory I
[ 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
Posted in Notes
Tagged basic, combinatorial game theory, computer science, impartial games, programming
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Symmetric Polynomials (III)
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