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Category Archives: Notes
Combinatorial Game Theory X
Lesson 10 In lesson 7, we learnt that if A, B are Left’s options in a game with A ≥ B, then we can drop B from the list of options and the game remains equal. In this lesson, we will … Continue reading
Combinatorial Game Theory IX
Lesson 9 Typically, at the end of a Domineering game, the board is divided into disjoint components, so the overall game is the (game) sum of the individual components. Suppose we have the following 6 components: How should the next … Continue reading
Combinatorial Game Theory VIII
Lesson 8 In this lesson, we will further familiarise ourselves with games involving numbers. At the end of the lesson, we will encounter our first positive infinitesimal: the “up” ↑. Here, an infinitesimal is a value which is strictly between –r and r … Continue reading
Combinatorial Game Theory VII
Lesson 7 [ Warning: another long post ahead. One of the proofs will also require mathematical induction. ] In this lesson, we will see how some games can be represented by numbers (which can be integers or fractions). We will also … Continue reading
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Tagged combinatorial game theory, computer science, game numbers, hackenbush, intermediate, partial games, programming
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Combinatorial Game Theory VI
Lesson 6 General Combinatorial Game Theory [ Warning: the following lesson is significantly longer than the previous ones. ] Starting from this lesson, we will look at a more rigourous, complete and general theory. Prior to this, in any game configuration both … Continue reading
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
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Tagged basic, combinatorial game theory, computer science, impartial games, loopy games, nim values, programming
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Combinatorial Game Theory IV
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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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
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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
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Tagged basic, combinatorial game theory, computer science, impartial games, programming
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Mathematics and Such 