Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Improving upper and lower bounds of the number of games born by day 4

Koki Suetsugu

Abstract

In combinatorial game theory, the lower and upper bounds of the number of games born by day $4$ have been recognized as $3.0 \cdot 10^{12}$ and $10^{434}$, respectively. In this study, we improve the lower bound to $10^{28.2}$ and the upper bound to $4.0 \cdot 10^{184}$, respectively.

Improving upper and lower bounds of the number of games born by day 4

Abstract

In combinatorial game theory, the lower and upper bounds of the number of games born by day have been recognized as and , respectively. In this study, we improve the lower bound to and the upper bound to , respectively.
Paper Structure (13 sections, 12 theorems, 8 equations, 4 figures, 6 tables, 1 algorithm)

This paper contains 13 sections, 12 theorems, 8 equations, 4 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Early results
  3. Division of $\mathbb{G}_3$
  4. Stratification of $\mathbb{G}_3$
  5. Chain division of $\mathbb{G}_3$
  6. Improving upper and lower bounds of $|\mathbb{G}_4|$
  7. Improving of lower bounds
  8. Further improvement of the lower bound
  9. Improving of the upper bound
  10. Further improvement of the upper bound
  11. Verification
  12. Conclusion
  13. $\mathbb{G}_3$ and its stratification

Key Result

Lemma 1

The stratification is upper and lower symmetric. That is, for any $u \in U_i$, $-u \in U_{46-i}$ holds.

Figures (4)

  • Figure 1: $22$ games born by day $2$ and their stratification
  • Figure 2: $1474$ games born by day $3$ and their stratification
  • Figure 3: $\mathcal{G}_3$
  • Figure 4: Chain division of games born by day $2$

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • Definition 1
  • Theorem 1
  • Lemma 3
  • Lemma 4
  • proof
  • Corollary 1
  • Corollary 2
  • Lemma 5
  • ...and 6 more