# The Search for the Longest Infinite Chess Game

TLDRThe video script delves into the theoretical exploration of infinite chess, where the traditional boundaries of the game are transcended with an infinite board. It discusses the concept of forced checkmate and the longest possible game lengths, introducing the audience to transfinite numbers like Omega and higher ordinals to quantify the game's duration. The script illustrates how black can prolong the game with 'announcements' and white must eventually force checkmate, even in positions with extraordinarily high game values. It concludes with the idea that while these game lengths are immense, they are still finite and do not reach the unattainable first uncountable ordinal, Omega 1.

###### Takeaways

- π€ The longest recorded chess game with perfect play is 8,848 moves, but it can be extended indefinitely by removing the 50-move rule and three-fold repetition.
- π² The concept of 'transfinite' chess introduces the idea of an infinite board and pieces, allowing for game lengths that exceed finite numbers.
- π³ The game tree in chess represents all possible moves and outcomes from a given position, with 'game value' decreasing towards zero for a forced checkmate.
- π In finite chess, white can force a checkmate in 549 moves, which is currently the longest known checkmate.
- π’ In the realm of infinite chess, the game value can be represented by transfinite ordinals, such as Omega, which is the smallest ordinal greater than all positive integers.
- π€ The value of a chess position in infinite chess cannot be calculated by simply adding one to the highest value below, as in finite chess, due to the approach to infinity.
- π Black has the power to control the length of the game in infinite chess by making strategic moves that delay the inevitable checkmate.
- π Higher ordinals, such as Omega squared or Omega to the power of Omega, represent positions in infinite chess where the game length is unimaginably long.
- ππ The concept of tiered announcements in infinite chess allows for a hierarchy of delays, with each tier adding another layer of complexity to the game length.
- π The search for the longest infinite chess game can theoretically continue indefinitely, with positions reaching far beyond Omega to the power of infinite ordinals.
- π« The upper bound for all things in infinite chess is Omega 1, the first uncountable ordinal, which represents a level of infinity that cannot be achieved in the game due to the countable nature of legal moves.

###### Q & A

### What is the significance of the 50-move rule and three-fold repetition in chess?

-The 50-move rule and three-fold repetition are mechanisms in chess that prevent the game from going on indefinitely without progress towards checkmate. Without these rules, players could make arbitrary moves indefinitely, leading to an endless game.

### What is the longest forced checkmate known in finite chess?

-The longest forced checkmate currently known in finite chess is 549 moves. This is achieved by discarding the 50-move rule and constructing a specific position where white can force checkmate through a series of moves.

### How does the concept of 'game tree' apply to chess?

-A game tree in chess illustrates all possible moves and outcomes from a given position. Each node represents a position, and each branch represents a possible move. The game tree helps visualize the progression of the game and the strategic decisions that players make.

### What is the concept of 'mate in Omega' in infinite chess?

-In infinite chess, 'mate in Omega' refers to a position where white has a forced checkmate, but the number of moves required exceeds all finite numbers. It represents a game length that is essentially unbounded, introducing the concept of transfinite values into the game.

### Why are ordinals used to represent the game value in infinite chess?

-Ordinals are used in infinite chess to represent game values that exceed finite numbers. They allow for the expression of positions where the game length can be incredibly long, such as 'mate in Omega', which cannot be accurately represented with finite numbers.

### What is the difference between 'mate in Omega' and 'mate in Omega squared'?

-While 'mate in Omega' indicates a game length that is unbounded but finite, 'mate in Omega squared' represents a position where black has the ability to make multiple announcements, each adding an Omega amount of delay to the game, resulting in a game length that is even more extensive.

### How can the game length be extended to 'Omega cubed' in infinite chess?

-In infinite chess, the game length can be extended to 'Omega cubed' by introducing a tier 2 announcement, which allows black to decide how many tier 1 announcements they receive. This creates a scenario where the number of announcements and the resulting game length can be incredibly vast.

### What is the significance of 'Epsilon 0' in the context of infinite chess?

-Epsilon 0 is a very large ordinal number that represents a game length in infinite chess that is beyond the comprehension of 'Omega to the power of Omega'. It is part of the process of continually increasing the game length by stacking ordinals.

### Why is 'Omega 1' considered the upper bound for all things in infinite chess?

-Omega 1, the first uncountable ordinal, is considered the upper bound for infinite chess because it represents a size that is uncountably infinite. Since chess positions can only have countably many legal moves at any given point, constructing a game with a value of 'mate in Omega 1' is impossible.

### What is the purpose of the Discord server mentioned in the script?

-The Discord server is a community space where players and enthusiasts of infinite chess can share their creations, discuss strategies, and collaborate on the development of new positions and variants of the game.

### How can one explore and create positions in infinite chess without an in-game board editor?

-In the absence of an in-game board editor, players can use a spreadsheet editor to create and explore positions in infinite chess. This involves copying and pasting configurations to simulate different game scenarios.

###### Outlines

##### π° The Quest for the Longest Chess Game

The script introduces the concept of the longest possible chess game, discussing the limitations of traditional chess rules and the exploration of 'Strange moves' that can lead to an endless game. It delves into the idea of perfect play and the longest forced checkmate known, which is 549 moves. The script then transitions into the realm of infinite chess with an expanded board, pondering the lengths of games that transcend finite numbers. The video promises a journey into the world of transfinites with the host, Navier, guiding the exploration of the longest infinite chess game.

##### π³ Understanding Game Trees and Infinite Chess

This paragraph explains the game tree concept, which illustrates all possible moves and outcomes from a given position. It describes how a perfect game would always decrease the game value by one with each optimal move, leading to checkmate at zero. The script simplifies the explanation by focusing on black's optimal strategy, which is to choose the move that maximizes the game value. It then introduces the complexities of infinite chess, where the game value can approach infinity, and the need for a new method to calculate game values, such as the smallest ordinal greater than all values below, exemplified by the concept of Omega.

##### π‘οΈ The Power of Ordinals in Infinite Chess

The script explores the use of transfinite ordinals as a measure for game length in infinite chess, where traditional numbers are insufficient. It discusses how black can control the game length by making strategic 'announcements' that determine how long the game will last. The concept of Omega is further expanded to include higher ordinals, such as Omega plus one, Omega squared, and beyond, showcasing positions where black has increasing stalling potential and flexibility. The research of mathematician Joel Hamkins and Professor Cory Evans is acknowledged for their contributions to the understanding of infinite chess.

##### π° The Complexity of Higher Ordinals in Chess

This section delves into the intricacies of higher ordinal values in chess positions, such as Omega cubed and tiered announcements, which allow black to decide the number of tier two announcements they receive. The script describes complex chess positions involving Rook Towers and Bishop Cannons, where each move and announcement adds layers of complexity and extends the game length exponentially. It illustrates how these positions can last an unimaginable number of moves, even when announcements are as small as 100, and how the game value is calculated by considering the smallest ordinal greater than all values below.

##### π Constructing Even Longer Infinite Games

The script introduces the concept of constructing even longer infinite chess games by adding nodes and protected squares, which increase the game length and create higher ordinal values. It explains how each node added can exponentially increase the game length, and how this method can be used to create positions with values of Omega squared or higher. The theoretical potential to reach higher and higher ordinals, such as Epsilon 0 and beyond, is discussed, highlighting the limitless nature of game construction in infinite chess.

##### π The Boundless Scale of Infinite Chess

This paragraph discusses the vast scale of infinite chess, where game lengths can reach beyond the comprehension of traditional numbers, entering the realm of transfinite ordinals. It explains the process of climbing the hierarchy of ordinals, from Omega to Epsilon, Zeta, and beyond, reaching the limits of Greek letters and entering the Veblen hierarchy. The script emphasizes the theoretical nature of these positions, acknowledging that while they may not be seen in competitive gameplay, they offer a fascinating exploration of the boundaries of chess and mathematics. The concept of Omega 1 as the upper bound for infinite chess is introduced, signifying the largest conceivable game length that remains unattainable.

##### π The Grand Vision for Infinite Chess

The final paragraph outlines the grand vision for infinite chess, including plans for infinite generation to bring many of the discussed positions to life and the creation of a devlog to showcase the problem-solving process behind the scenes. The script invites viewers to join a Discord server to share their creations and participate in the community, expressing gratitude to patrons and researchers for their contributions to the understanding and exploration of infinite chess. It concludes with a teaser for future content and a farewell until the next video.

###### Mindmap

###### Keywords

##### π‘Chess

##### π‘Transfinite Ordinals

##### π‘Checkmate

##### π‘Game Tree

##### π‘Infinite Chess

##### π‘50-Move Rule

##### π‘Three-Fold Repetition

##### π‘Perfect Play

##### π‘Zugzwang

##### π‘Epsilon 0

##### π‘Omega 1

###### Highlights

The longest possible chess game is 8,848 moves without the 50-move rule and three-fold repetition.

A position is described where white can force checkmate in 549 moves, the longest known in finite chess.

Infinite chess on a larger board can surpass finite numbers for the checkmate clock.

The game tree concept is introduced to understand all possible moves and outcomes from a given position.

White's strategy in a finite chess game is to always reduce the game value by one with optimal play.

Calculating the value of a node in chess involves looking at all possible moves and their outcomes.

Infinite chess requires a new approach to calculate game values, as numbers approach infinity.

A position claimed to be mate in Omega, the first infinite ordinal, is introduced.

Omega is defined as the smallest ordinal greater than all positive integers.

The concept of 'announcements' in infinite chess is introduced, where black controls the game length.

Higher ordinals like Omega plus one, Omega squared, and beyond are explored for infinite chess positions.

Positions with game values of Omega to the power of various ordinals are constructed.

The concept of tiered announcements in infinite chess is explained, allowing for even longer game lengths.

Epsilon0, a much larger ordinal, is introduced as a potential game value in infinite chess.

The Velin hierarchy and Greek letter ordinals are used to describe even higher levels of infinite game values.

Omega1, the first uncountable ordinal, is identified as the upper bound for infinite chess game lengths.

Infinite chess is framed as a game for the gods, with lengths too long for mortal play.

The video discusses future plans for infinite chess, including infinite generation and a devlog.

A spreadsheet editor for infinite chess positions is made available to the community through a Discord server.

###### Transcripts

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