Pre-Grundy Games Games And Graphs Workshop 2017 In collaboration - PowerPoint PPT Presentation

Octal Games Pre-Grundy Games thks Pre-Grundy Games Games And Graphs Workshop 2017 In collaboration with : ´ Eric Duchˆ ene, Antoine Dailly and Urban Larsson Gabrielle Paris 1/26

Octal Games Pre-Grundy Games thks Are Pre-Grundy games boring ? 2/26

Octal Games Pre-Grundy Games thks Sommaire Octal Games 1 Pre-Grundy Games 2 3/26

Octal Games Pre-Grundy Games thks Definition Octal games: [Winning Ways] A game played on heaps where each player: 1. 2. 3. 4/26

Octal Games Pre-Grundy Games thks Definition Octal games: [Winning Ways] A game played on heaps where each player: 1. removes all the tokens of a heap, or 1. 2. 3. 4/26

Octal Games Pre-Grundy Games thks Definition Octal games: [Winning Ways] A game played on heaps where each player: 1. removes all the tokens of a heap, or 2. removes only some, at the end, or 1. 2. 3. 4/26

Octal Games Pre-Grundy Games thks Definition Octal games: [Winning Ways] A game played on heaps where each player: 1. removes all the tokens of a heap, or 2. removes only some, at the end, or 3. removes only some in the middle 1. 2. 3. 4/26

Octal Games Pre-Grundy Games thks Definition Octal games: [Winning Ways] A game played on heaps where each player: 1. removes all the tokens of a heap, or 2. removes only some, at the end, or 3. removes only some in the middle Coded by an octal number 0 . d 1 . . . d t 1. 2. 3. 4/26

Octal Games Pre-Grundy Games thks Example of octal games • Nim: 0 . 33333333333333333 ... 1. 2. 3. 5/26

Octal Games Pre-Grundy Games thks Example of octal games • Nim: 0 . 33333333333333333 ... 1. 2. 3. 5/26

Octal Games Pre-Grundy Games thks Example of octal games • Nim: 0 . 33333333333333333 ... • Kayles: 0 . 77 . 1. 2. 3. 5/26

Octal Games Pre-Grundy Games thks Example of octal games • Nim: 0 . 33333333333333333 ... • Kayles: 0 . 77 . 1. 2. 3. 5/26

Octal Games Pre-Grundy Games thks Example of octal games • Nim: 0 . 33333333333333333 ... • Kayles: 0 . 77 . • Dawson Chess: 0 . 137 . 1. 2. 3. 5/26

Octal Games Pre-Grundy Games thks Example of octal games • Nim: 0 . 33333333333333333 ... • Kayles: 0 . 77 . • Dawson Chess: 0 . 137 . 1. 2. 3. 5/26

Octal Games Pre-Grundy Games thks Example of octal games • Nim: 0 . 33333333333333333 ... • Kayles: 0 . 77 . • Dawson Chess: 0 . 137 . 1. 2. 3. Conjecture (Guy) Octal games with finite code have a Grundy sequence ultimately periodic. 5/26

Octal Games Pre-Grundy Games thks Hexadecimal games Why stop at two heaps ? 4. 6/26

Octal Games Pre-Grundy Games thks Hexadecimal games Why stop at two heaps ? 4. 5. ⇒ we can simply generalize to leaving any number of heaps... 6/26

Octal Games Pre-Grundy Games thks Grundy sequences Game 0 . F2 : periodic 15 10 5 0 0 5 10 15 20 7/26

Octal Games Pre-Grundy Games thks Grundy sequences Game 0 . 17FF : arithmetic-periodic 15 10 5 0 0 5 10 15 20 7/26

Octal Games Pre-Grundy Games thks Periodicity and arithmetic-periodicity results Theorem (WW) Let H be the hexadecimal game 0 . d 1 . . . d t . If there exist e and p such that: ∀ e < n ≤ 3( e + p ) + t , G ( n + p ) = G ( n ) then the Grundy sequence of H is periodic of period p and with pre-period e. 8/26

Octal Games Pre-Grundy Games thks Periodicity and arithmetic-periodicity results Theorem (WW) Let H be the hexadecimal game 0 . d 1 . . . d t . If there exist e and p such that: ∀ e < n ≤ 3( e + p ) + t , G ( n + p ) = G ( n ) then the Grundy sequence of H is periodic of period p and with pre-period e. Theorem (R. Nowakowski) Let H be the hexadecimal game 0 . d 1 . . . d t . If there exist e, 3 p ≥ t + 2 , s = 2 γ − 1 + j, j < 2 γ − 1 such that: 1. ∀ e < n < t + α e ,γ, j p , G ( n + p ) = G ( n ) + s , 2. G ( � 0 , e � ) ⊂ � 0 , s − 1 � and G ( � 0 , e + p � ) ⊂ � 0 , 2s − 1 � , 3. ∃ d 2 v +1 , d v ≥ 8 , ∀ g ∈ � 0 , 2s − 1 � , ∃ n > 0 , G ( n ) = g or ∃ d u ≥ 8 , ∀ g ∈ � 0 , 2s − 1 � , ∃ 2 v + 1 , 2 w ≥ 0 , G (2 v + 1) = G (2 w ) = g , then the Grundy sequence of H is arithmetic-periodic of period p, pre-period e and saltus s. 8/26

Octal Games Pre-Grundy Games thks Periodicity and arithmetic-periodicity results Theorem (WW) Let H be the hexadecimal game 0 . d 1 . . . d t . If there exist e and p such that: ∀ e < n ≤ 3( e + p ) + t , G ( n + p ) = G ( n ) then the Grundy sequence of H is periodic of period p and with pre-period e. Theorem (R. Nowakowski) Let H be the hexadecimal game 0 . d 1 . . . d t . If there exist e, 3 p ≥ t + 2 , s = 2 γ − 1 + j, j < 2 γ − 1 such that: 1. ∀ e < n < t + α e ,γ, j p , G ( n + p ) = G ( n ) + s , 2. G ( � 0 , e � ) ⊂ � 0 , s − 1 � and G ( � 0 , e + p � ) ⊂ � 0 , 2s − 1 � , 3. ∃ d 2 v +1 , d v ≥ 8 , ∀ g ∈ � 0 , 2s − 1 � , ∃ n > 0 , G ( n ) = g or ∃ d u ≥ 8 , ∀ g ∈ � 0 , 2s − 1 � , ∃ 2 v + 1 , 2 w ≥ 0 , G (2 v + 1) = G (2 w ) = g , then the Grundy sequence of H is arithmetic-periodic of period p, pre-period e and saltus s. 8/26

Octal Games Pre-Grundy Games thks Sommaire Octal Games 1 Pre-Grundy Games 2 9/26

Octal Games Pre-Grundy Games thks Grundy’s Game A move consists in taking a heap and splitting it into two non-empty heaps of different sizes. No removing allowed. 1 1 10/26

Octal Games Pre-Grundy Games thks Grundy’s Game A move consists in taking a heap and splitting it into two non-empty heaps of different sizes. No removing allowed. 1 1 Conjecture (Belekamp, Conway, Guy) The Grundy sequence of Grundy’s game is ultimately periodic. Remark: 2 35 first values computed (Flammenkamp) without any further clue... 10/26

Octal Games Pre-Grundy Games thks Pre-Grundy Games Definition Let L = { ℓ 1 , . . . , ℓ k } be a list of positive integers. A move on the game PreG( L ) consist on splitting a heap of n ≥ ℓ j tokens into ℓ j + 1 non-empty heaps. L = { 1 , 3 } 1 1 2 3 11/26

Octal Games Pre-Grundy Games thks Behavior L = { 1 , 3 } : P 3 p = 2 0 0 5 10 15 20 L = { 1 , 3 , 4 } : AP 9 p = 4; s = 2 5 0 0 5 10 15 20 12/26

Octal Games Pre-Grundy Games thks Firsts results L = { ℓ 1 , . . . , ℓ k } type Sequence (0) ℓ 1 (+1) 1 / ∈ L AP Proofs: For L = { ℓ 1 , . . . , ℓ k } , ℓ i > 1. For n = a ℓ 1 + b + 1, b < ℓ 1 , let us prove that G ( n ) = a . First: G ( n ) ≥ a . 13/26

Octal Games Pre-Grundy Games thks Firsts results L = { ℓ 1 , . . . , ℓ k } type Sequence (0) ℓ 1 (+1) 1 / ∈ L AP Proofs: For L = { ℓ 1 , . . . , ℓ k } , ℓ i > 1. For n = a ℓ 1 + b + 1, b < ℓ 1 , let us prove that G ( n ) = a . First: G ( n ) ≥ a . • ℓ 1 even: O ℓ 1 = ( i ℓ 1 + b + 1 , a − i , . . . , a − i ) G ( O ℓ 1 ) = G ( i ℓ 1 + b + 1) = i 13/26

Octal Games Pre-Grundy Games thks Firsts results L = { ℓ 1 , . . . , ℓ k } type Sequence (0) ℓ 1 (+1) 1 / ∈ L AP Proofs: For L = { ℓ 1 , . . . , ℓ k } , ℓ i > 1. For n = a ℓ 1 + b + 1, b < ℓ 1 , let us prove that G ( n ) = a . First: G ( n ) ≥ a . • ℓ 1 odd: − if a − i odd: � i ℓ 1 + b + 1 , 1 , a − i − 1 ℓ 1 + 1 , a − i − 1 � O ℓ 1 = ℓ 1 + 1 , 1 , . . . , 1 2 2 G ( O ℓ 1 ) = G ( i ℓ 1 + b + 1) ⊕ G (1) = i 13/26

Octal Games Pre-Grundy Games thks Firsts results L = { ℓ 1 , . . . , ℓ k } type Sequence (0) ℓ 1 (+1) 1 / ∈ L AP Proofs: For L = { ℓ 1 , . . . , ℓ k } , ℓ i > 1. For n = a ℓ 1 + b + 1, b < ℓ 1 , let us prove that G ( n ) = a . First: G ( n ) ≥ a . • ℓ 1 odd: − if a − i even: � i ℓ 1 + b + 1 , 2 , a − i − 1 ℓ 1 + 1 2 , a − i − 1 ℓ 1 + 1 � O ℓ 1 = 2 , 1 , . . . , 1 2 2 G ( O ℓ 1 ) = G ( i ℓ 1 + b + 1) ⊕ G (2) = i 13/26

Octal Games Pre-Grundy Games thks Firsts results L = { ℓ 1 , . . . , ℓ k } type Sequence (0) ℓ 1 (+1) 1 / ∈ L AP Proofs: For L = { ℓ 1 , . . . , ℓ k } , ℓ i > 1. For n = a ℓ 1 + b + 1, b < ℓ 1 , let us prove that G ( n ) = a . First: G ( n ) ≥ a . In all cases G ( O ℓ 1 ) = i for i < a ⇒ G ( n ) ≥ a . 13/26

Octal Games Pre-Grundy Games thks Firsts results L = { ℓ 1 , . . . , ℓ k } type Sequence (0) ℓ 1 (+1) 1 / ∈ L AP Proofs: For L = { ℓ 1 , . . . , ℓ k } , ℓ i > 1. For n = a ℓ 1 + b + 1, b < ℓ 1 , let us prove that G ( n ) = a . Second: G ( n ) ≤ a . Assume G ( O ℓ ) = a for some option of n . 14/26

Octal Games Pre-Grundy Games thks Firsts results L = { ℓ 1 , . . . , ℓ k } type Sequence (0) ℓ 1 (+1) 1 / ∈ L AP Proofs: For L = { ℓ 1 , . . . , ℓ k } , ℓ i > 1. For n = a ℓ 1 + b + 1, b < ℓ 1 , let us prove that G ( n ) = a . Second: G ( n ) ≤ a . Assume G ( O ℓ ) = a for some option of n . O ℓ = ( a 0 ℓ 1 + b 0 + 1 , . . . , a ℓ ℓ 1 + b ℓ + 1) 14/26

Octal Games Pre-Grundy Games thks Firsts results L = { ℓ 1 , . . . , ℓ k } type Sequence (0) ℓ 1 (+1) 1 / ∈ L AP Proofs: For L = { ℓ 1 , . . . , ℓ k } , ℓ i > 1. For n = a ℓ 1 + b + 1, b < ℓ 1 , let us prove that G ( n ) = a . Second: G ( n ) ≤ a . Assume G ( O ℓ ) = a for some option of n . O ℓ = ( a 0 ℓ 1 + b 0 + 1 , . . . , a ℓ ℓ 1 + b ℓ + 1) � a i = a • G ( O ℓ ) = a ⇒ 14/26