Octal Games on Graphs Laurent Beaudou1, Pierre Coupechoux2, Antoine Dailly3, Sylvain Gravier4, Julien Moncel2, Aline Parreau3, Éric Sopena5 1LIMOS, Clermont-Ferrand 2LAAS, Toulouse 3LIRIS, Lyon 4Institut Fourier, Grenoble 5LaBRI, Bordeaux This work is part of the ANR GAG (Graphs and Games). Games And Graphs CGTC 2017 1/19 I kayles is 0.137 I cram on a single row is 0.07 I The James Bond Game is 0.007 I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. Octal games Definition Octal games are: 2/19 I kayles is 0.137 I cram on a single row is 0.07 I The James Bond Game is 0.007 Examples I nim is 0.3333. Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. 2/19 I kayles is 0.137 I cram on a single row is 0.07 I The James Bond Game is 0.007 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. 2/19 I cram on a single row is 0.07 I The James Bond Game is 0.007 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 2/19 I cram on a single row is 0.07 I The James Bond Game is 0.007 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 2/19 I cram on a single row is 0.07 I The James Bond Game is 0.007 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 2/19 I cram on a single row is 0.07 I The James Bond Game is 0.007 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 2/19 I cram on a single row is 0.07 I The James Bond Game is 0.007 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 2/19 I cram on a single row is 0.07 I The James Bond Game is 0.007 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 2/19 I The James Bond Game is 0.007 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 I cram on a single row is 0.07 2/19 I The James Bond Game is 0.007 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 I cram on a single row is 0.07 2/19 I The James Bond Game is 0.007 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 I cram on a single row is 0.07 2/19 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 I cram on a single row is 0.07 I The James Bond Game is 0.007 2/19 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 I cram on a single row is 0.07 I The James Bond Game is 0.007 2/19 Octal games Definition Octal games are: I impartial games; I played on heaps of counters; I whose rules are defined by an octal code. Examples I nim is 0.3333. I kayles is 0.137 I cram on a single row is 0.07 I The James Bond Game is 0.007 2/19 I kayles: 0,1,2,3,1,4,3,2,. after a pre-period 72 it becomes periodic with period 12; I cram on a single row: 0,1,1,2,0,3,1,1,. after a pre-period 53 it becomes periodic with period 34 I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. still open,2 28 values computed! Examples I nim: 0,1,2,3,4,5,. Conjecture (Guy) All finite octal games have ultimately periodic Grundy sequences. Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0; 1; 2;::: 3/19 I kayles: 0,1,2,3,1,4,3,2,. after a pre-period 72 it becomes periodic with period 12; I cram on a single row: 0,1,1,2,0,3,1,1,. after a pre-period 53 it becomes periodic with period 34 I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. still open,2 28 values computed! Conjecture (Guy) All finite octal games have ultimately periodic Grundy sequences. Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0; 1; 2;::: Examples I nim: 0,1,2,3,4,5,. 3/19 I cram on a single row: 0,1,1,2,0,3,1,1,. after a pre-period 53 it becomes periodic with period 34 I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. still open,2 28 values computed! Conjecture (Guy) All finite octal games have ultimately periodic Grundy sequences. Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0; 1; 2;::: Examples I nim: 0,1,2,3,4,5,. I kayles: 0,1,2,3,1,4,3,2,. after a pre-period 72 it becomes periodic with period 12; 3/19 I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. still open,2 28 values computed! Conjecture (Guy) All finite octal games have ultimately periodic Grundy sequences. Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0; 1; 2;::: Examples I nim: 0,1,2,3,4,5,. I kayles: 0,1,2,3,1,4,3,2,. after a pre-period 72 it becomes periodic with period 12; I cram on a single row: 0,1,1,2,0,3,1,1,. after a pre-period 53 it becomes periodic with period 34 3/19 Conjecture (Guy) All finite octal games have ultimately periodic Grundy sequences. Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0; 1; 2;::: Examples I nim: 0,1,2,3,4,5,. I kayles: 0,1,2,3,1,4,3,2,. after a pre-period 72 it becomes periodic with period 12; I cram on a single row: 0,1,1,2,0,3,1,1,. after a pre-period 53 it becomes periodic with period 34 I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. still open,2 28 values computed! 3/19 Octal games Grundy sequence The Grundy sequence of an octal game is the sequence of its Grundy values for heaps of size 0; 1; 2;::: Examples I nim: 0,1,2,3,4,5,. I kayles: 0,1,2,3,1,4,3,2,. after a pre-period 72 it becomes periodic with period 12; I cram on a single row: 0,1,1,2,0,3,1,1,. after a pre-period 53 it becomes periodic with period 34 I The James Bond Game: 0,0,0,1,1,1,2,2,0,3,3,1,1,1,0,4,. still open,2 28 values computed! Conjecture (Guy) All finite octal games have ultimately periodic Grundy sequences. 3/19 Removing connected vertices Removing counters from a heap from a graph Splitting a heap Disconnecting a graph Playing on a heap ≡ Playing on a path Octal games on graphs Natural generalization of the definition: Playing on heaps Playing on graphs 4/19 Removing connected vertices from a graph Splitting a heap Disconnecting a graph Playing on a heap ≡ Playing on a path Octal games on graphs Natural generalization of the definition: Playing on heaps Playing on graphs Removing counters from a heap 4/19 Removing connected vertices from a graph Splitting a heap Disconnecting a graph Playing on a heap ≡ Playing on a path Octal games on graphs Natural generalization of the definition: Playing on heaps Playing on graphs Removing counters from a heap 4/19 Splitting a heap Disconnecting a graph Playing on a heap ≡ Playing on a path Octal games on graphs Natural generalization of the definition: Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph 4/19 Splitting a heap Disconnecting a graph Playing on a heap ≡ Playing on a path Octal games on graphs Natural generalization of the definition: Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph 4/19 Disconnecting a graph Playing on a heap ≡ Playing on a path Octal games on graphs Natural generalization of the definition: Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph Splitting a heap 4/19 Disconnecting a graph Playing on a heap ≡ Playing on a path Octal games on graphs Natural generalization of the definition: Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph Splitting a heap 4/19 Disconnecting a graph Playing on a heap ≡ Playing on a path Octal games on graphs Natural generalization of the definition: Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph Splitting a heap 4/19 Playing on a heap ≡ Playing on a path Octal games on graphs Natural generalization of the definition: Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph Splitting a heap Disconnecting a graph 4/19 Playing on a heap ≡ Playing on a path Octal games on graphs Natural generalization of the definition: Playing on heaps Playing on graphs Removing connected vertices Removing counters from a heap from a graph Splitting a heap Disconnecting