Author Archives: limsup

Combinatorial Game Theory VII

Posted on August 15, 2012 by limsup

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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Combinatorial Game Theory VI

Posted on August 12, 2012 by limsup

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 →

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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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Thinking Infinitesimally – Multivariate Calculus (II)

Posted on April 17, 2012 by limsup

Chain Rule for Multivariate Calculus We continue our discussion of multivariate calculus. The first item here is the analogue of Chain Rule for the multivariate case. Suppose we have parameters f, u, v, x, y, z. Suppose {u, v} are independent parameters (in particular, … Continue reading →

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Thinking Infinitesimally – Multivariate Calculus (I)

Posted on April 15, 2012 by limsup

[ Background required: some understanding of single-variable calculus, including differentiation and integration. ] The object of this series of articles is to provide a rather different point-of-view to multivariate calculus, compared to the conventional approach in calculus texts. The typical … Continue reading →

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