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Monthly Archives: August 2012
Combinatorial Game Theory XII
Lesson 12 Recall the following Domineering configuration in lesson 10: The above game has a nice theory behind it. Definition : For any game G, the game –G (called miny–G) is defined to be: The game +G (called tiny–G) is defined … Continue reading
Posted in Notes
Tagged combinatorial game theory, computer science, domineering, game infinitesimals, impartial games, intermediate, minies, tinies, toads and frogs
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Combinatorial Game Theory XI
Lesson 11 In this lesson, we will cover more on canonical forms. First recall that for m↑ + (*n) with m > 0, this game is positive except when (m, n) = (1, 1). Let’s consider the canonical forms of … Continue reading
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 Quiz 2
This quiz lasts 70 minutes and covers materials from lessons 19. Use the simplicity rule to compute the values of the following games. (10 points) {1/2  } {1/4  } {1/8  3/8} {0  7/8} {1/8  9/16} … Continue reading
Posted in Homework
Tagged combinatorial game theory, computer science, game numbers, intermediate, partial games, up game
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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
Posted in Notes
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
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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