Impartial Games for Generating Groups

Loosely speaking, a group is a set together with an associative binary operation that satisfies a few modest conditions: the "product" of any two elements from the set is an element of the set (closure), there exists a "do nothing" element (identity), and for every element in the set, there exists another element in the set that "undoes" the original (inverses). Let G be a finite group. Given a single element from G, we can create new elements of the group by raising the element to various powers. Given two elements, we have even more options for creating new elements by combining powers of the two elements. Since G is finite, some finite number of elements will "generate" all of G. In the game DO GENERATE, two players alternately select elements from G. At each stage, a group is generated by the previously selected elements. The winner is the player that generates all of G. There is an alternate version of the game called DO NOT GENERATE in which the loser is the player that generates all of G. In this talk, we will explore both games and discuss winning strategies. Time permitting, we may also relay some current research related to both games.

This talk was given on April 23, 2013 as part of the Cool Math Talk series at the University of Nebraska at Omaha.

Transcript

1. Impartial games for generating groups
   Dana C. Ernst
   Northern Arizona University
   Department of Mathematics and Statistics
   http://danaernst.com
   UNO Cool Math Talk Series
   April 23, 2013
   D.C. Ernst Impartial games for generating groups 1 / 21
   

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2. Combinatorial Game Theory
   Intuitive Definition
   Combinatorial Game Theory (CGT) is the study of two-person games satisfying:
   • Two players alternate making moves.
   • No hidden information.
   • No random moves.
   • The game sequence is finite and there are no ties.
   • Normal Play: The last play to move wins.
   • Mis`
   ere Play: The last player to move loses.
   Example
   Combinatorial games:
   • Chess
   • Connect Four
   • Nim
   • X-Only Tic-Tac-Toe
   Non-combinatorial games:
   • Battleship (hidden information)
   • Rock-Paper-Scissors (non-alternating and random)
   • Poker (hidden information and random)
   • Tic-Tac-Toe (ties are possible)
   D.C. Ernst Impartial games for generating groups 2 / 21
   

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3. Impartial vs Partizan
   Definition
   A combinatorial game is called impartial if the move options are the same for both
   players. Otherwise, the game is called partizan.
   Example
   Partizan:
   • Chess
   • Connect Four
   Impartial:
   • Nim
   • X-Only Tic-Tac-Toe
   Note
   • We will explore a few impartial games.
   • When analyzing games, we will assume that both players make optimal moves.
   • The player that moves first is called α and the second player is called β.
   D.C. Ernst Impartial games for generating groups 3 / 21
   

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4. Nim
   Single-pile Nim
   Start with a pile of n stones. Each player chooses at least
   one stone from the pile. The player that takes the last
   stone wins.
   .
   .
   .
   .
   .
   .
   n
   Question
   Is there an optimal strategy for either player?
   Answer
   This game is sort of boring as α always wins; just take the whole pile.
   Let’s crank it up a notch.
   Multi-pile Nim
   Start with k piles of stones consisting of n1, n2, . . . , nk
   stones, respectively. Each
   player chooses at least one stone from a single pile. The player that takes the last
   stone wins.
   D.C. Ernst Impartial games for generating groups 4 / 21
   

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5. Nim (continued)
   Example
   Let’s start with 3 piles consisting of 1, 2, and 3 stones. Here’s a possible sequence.
   .
   . (1, 2, 3)
   α
   →
   .
   . (1, 2, 2)
   β
   →
   .
   . (0, 2, 2)
   α
   →
   .
   . (0, 1, 2)
   β
   →
   .
   . (0, 1, 1)
   α
   →
   .
   . (0, 1, 0)
   β
   →
   .
   .
   Yay!
   . (0, 0, 0)
   In this case, β wins.
   Question
   Is there an optimal strategy for either player?
   Answer
   The short answer is yes. The big picture is to whittle down to an even number of
   piles with a single stone. If the players make optimal moves, this is only possible for
   one of the players. The long answer involves nimbers.
   D.C. Ernst Impartial games for generating groups 5 / 21
   

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6. Mis`
   ere Nim
   Mis`
   ere Version of Nim
   In this alternate version of Nim, the player that takes the last stone loses.
   Example
   Let’s start with (1, 2, 3) again.
   .
   . (1, 2, 3)
   α
   →
   .
   . (1, 2, 2)
   β
   →
   .
   . (0, 2, 2)
   α
   →
   .
   . (0, 1, 2)
   β
   →
   .
   . (0, 1, 0)
   α
   →
   .
   .
   Doh!
   . (0, 0, 0)
   β wins again.
   Optimal Play
   • If there is a single pile, then α wins by taking all but one stone.
   • If there is more than one pile, then the general strategy is to whittle down to an
   odd number of piles with a single stone.
   • As with normal-play Nim, a more detailed analysis involves nimbers.
   D.C. Ernst Impartial games for generating groups 6 / 21
   

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7. X-Only Tic-Tac-Toe
   Single Board X-Only Tic-Tac-Toe
   Start with a single ordinary Tic-Tac-Toe board. Place a single X in any empty
   square. The first player to get 3 in a row wins.
   Example
   Let’s take a look at an example.
   α
   →
   X
   β
   →
   X
   X
   α
   →
   X
   X
   X
   β
   →
   X
   X
   X X
   α
   →
   X X
   X
   X X
   Boom, α wins.
   Optimal Play
   It’s pretty easy to see that α can always wins. In fact, if α plays in the middle square,
   the game is over quickly.
   D.C. Ernst Impartial games for generating groups 7 / 21
   

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8. Mis`
   ere X-Only Tic-Tac-Toe
   Single-board Notakto
   In the mis`
   ere version of X-Only Tic-Tac-Toe (also called Notakto), the player that
   gets three in a row loses.
   Question
   Is there are clear winner?
   Optimal Play
   If α’s first move is in the center square, then α wins.
   X
   Question
   What happens if we increase the number of boards that we can play on?
   D.C. Ernst Impartial games for generating groups 8 / 21
   

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9. Multi-Board X-Only Tic-Tac-Toe
   Multi-Board X-Only Tic-Tac-Toe
   Suppose there are k boards.
   • Normal Play: Players place an X in any open square on a single board. Once a
   board has 3 in row, that board is removed from play. The player that gets 3 in a
   row on the last remaining board wins.
   • Notakto: In the mis`
   ere version, the player to get three in a row on the last
   remaining board loses.
   Optimal Play
   For Notakto, a complete analysis involves an 18 element commutative monoid.
   Note
   • If you want to know more, check out “The Secrets of Notakto: Winning at
   X-only Tic-Tac-Toe” (http://arxiv.org/abs/1301.1672).
   • Also, check out the free iPad app called Notakto.
   D.C. Ernst Impartial games for generating groups 9 / 21
   

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10. Groups
    Before discussing the next game, we need to introduce groups.
    Groups are fundamental objects in mathematics. Loosely speaking, start with a
    collection of objects, throw in a method for combining two objects together so that it
    satisfies some reasonable requirements and you’ve got yourself a group.
    Intuitive Definition
    Slightly more rigorously, a group is a set with an associative binary operation
    satisfying:
    • Closure: “Product” of any two elements from the set is an element of the set.
    • Identity: There exists a “do nothing” element.
    • Inverses: For every element in the set, there exists another element in the set
    that “undoes” the original.
    D.C. Ernst Impartial games for generating groups 10 / 21
    

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11. Examples of Groups
    Example
    • Z is a group under addition. Identity is 0 and inverse of n is −n.
    • Z under multiplication is not a group. Why? If n ̸= ±1, then the inverse of n is
    not an integer.
    • R \ {0} is a group under multiplication. Identity is 1 and inverse of each
    non-zero real number is its reciprocal.
    • The set Zn
    = {0, 1, 2, . . . , n − 1} is a group under addition modulo n. Identity is
    0 and inverse of k is n − k.
    • The set Dn
    of symmetries (rotations and reflections) of a regular n-gon is a
    group under composition. Identity is the rotation by 0◦, inverse of a rotation is
    the rotation in the opposite direction, and inverse of a reflection is the same
    reflection. Dn
    is non-commutative (i.e., order of composition matters) and
    consists of 2n elements.
    • The set Sn
    of permutations of n objects under composition is a
    non-commutative group with n! elements. Identity is the element that does not
    scramble anything and the inverse of a permutation is the permutation that
    reverses the scrambling.
    D.C. Ernst Impartial games for generating groups 11 / 21
    

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12. Group Tables
    One way of representing a finite group is with a group table.
    Example
    The following table depicts Z6
    (written multiplicatively).
    The product of x times y is the entry
    in the row labelled x and column
    labelled y.
    D.C. Ernst Impartial games for generating groups 12 / 21
    

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13. Group Tables (continued)
    Example
    The following table depicts D4
    (symmetries of a square). In this case, r is rotation by
    90◦ clockwise, f is reflection across one of the diagonals, and e is the identity.
    Observations
    • Order matters, but not always
    (e.g., rf ̸= fr, fr2 = r2f ).
    • Every element has been written
    in terms of r and f .
    • e, r, r2, r3 are rotations.
    • f , rf , fr, r2f are reflections.
    • Composition of two reflections is
    a rotation (by twice the angle
    between them).
    D.C. Ernst Impartial games for generating groups 13 / 21
    

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14. Generators
    Definition
    Let G be a finite group. The set generated by g1, g2, . . . , gn ∈ G is the subset of
    elements in G that we can construct using only g1, g2, . . . , gn
    . We denote this set via
    ⟨g1, g2, . . . , gn⟩.
    Example
    Consider Z6
    . Then
    • ⟨0⟩ = {0}
    • ⟨2⟩ = {2, 4, 0}
    • ⟨2, 4⟩ = {2, 4, 0}
    • ⟨3⟩ = {3, 0}
    • ⟨2, 3⟩ = {2, 4, 0, 3, 5, 1} = Z6
    • ⟨1⟩ = {1, 2, 3, 4, 5, 0} = Z6
    Consider D4
    . Then
    • ⟨e⟩ = {e}
    • ⟨f ⟩ = {f , e}
    • ⟨r⟩ = {r, r2, r3, e}
    • ⟨r2⟩ = {r2, e}
    • ⟨r, f ⟩ = {r, r2, r3, e, f , fr, rf , r2f } = D4
    • ⟨f , fr⟩ = {f , e, fr, r, r2, r3, rf , r2f } = D4
    D.C. Ernst Impartial games for generating groups 14 / 21
    

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15. Generators (continued)
    Fact 1
    If G is a finite group, then there is always a finite set of elements that generates all
    of G.
    Fact 2
    We always have ⟨g1, g2, . . . , gk
    ⟩ ⊆ ⟨g1, g2, . . . , gk
    , gk+1
    ⟩ (and may have equality).
    Fact 3
    Zn
    = ⟨k⟩ if and only if n and k are relatively prime (i.e., have no prime factors in
    common).
    Fact 4
    Dn
    = ⟨rk , f ⟩ = ⟨f , f ′⟩, where r is a single-click rotation, k is relatively prime to n, and
    f and f ′ are any adjacent reflections. No single element generates Dn
    (for n ≥ 3).
    Careful!
    There may be larger sets that generate these groups having the property that no
    strictly smaller subset generates the whole group (e.g., ⟨2, 3⟩ = Z6
    ).
    D.C. Ernst Impartial games for generating groups 15 / 21
    

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16. GENERATE Game
    The following game was introduced by F. Harary in 1987.
    GENERATE Game
    Let G be a finite group with more than one element. On the first move, α chooses
    some g1 ∈ G. On the kth move, a player chooses gk
    ∈ G \ {g1, g2, . . . gk−1
    }. The
    winner is the player that generates all of G with their choice together with previous
    choices.
    Choice Set Generated
    g1 ⟨g1⟩
    g2 ⟨g1, g2⟩
    g3 ⟨g1, g2, g3⟩
    .
    .
    .
    .
    .
    .
    gk−1
    ⟨g1, g2, g3, . . . gk−1
    ⟩ ̸= G
    gk
    ⟨g1, g2, g3, . . . gk−1
    , gk
    ⟩ = G
    The player that chooses gk
    wins.
    D.C. Ernst Impartial games for generating groups 16 / 21
    

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17. GENERATE for Zn
    Example
    GENERATE is pretty boring with Zn
    .
    Let’s look at Z6
    (with non-optimal play).
    Choice Set Generated
    a4 ⟨a4⟩ = {a4, a2, e}
    a3 ⟨a4, a3⟩ = {a4, a2, e, a3, a, a5}
    In this case, β wins. However, α could
    have won immediately by choosing a or a5.
    Optimal play for Zn
    For Zn
    , α can always win GENERATE in one move by choosing any ak , where n and
    k are relatively prime.
    D.C. Ernst Impartial games for generating groups 17 / 21
    

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18. GENERATE for Dn
    Example
    Dn
    is more interesting. Let’s look at D4
    .
    Choice Set Generated
    r2 ⟨r2⟩ = {r2, e}
    e ⟨r2, e⟩ = {r2, e}
    f ⟨r2, e, f ⟩ = {r2, e, f , r2f }
    r ⟨r2, e, f , r⟩ = D4
    β wins again. In fact, we have the
    following fact.
    Optimal play for Dn
    with n a multiple of 4
    In this case, β is guaranteed to win (F.W. Barnes, 1988). General strategy: If α
    chooses rk with n and k relatively prime or any reflection, then β wins on next move.
    To win, β stalls by picking available rm with n and m not relatively prime.
    D.C. Ernst Impartial games for generating groups 18 / 21
    

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19. GENERATE for Dn (continued)
    Example
    What if n is not a multiple of 4? Let’s
    look at D5
    .
    Choice Set Generated
    e ⟨e⟩ = {e}
    r2 ⟨r2⟩ = {r2, frf = r4, r, r3, e}
    f D5
    This time α wins. And when n is not a
    multiple of 4, α can always win.
    Optimal play for Dn
    with n not a multiple of 4
    In this case, α is guaranteed to win (F.W. Barnes, 1988).
    D.C. Ernst Impartial games for generating groups 19 / 21
    

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20. DO NOT GENERATE Game
    As with the other games, there is a mis`
    ere version of GENERATE.
    DO NOT GENERATE Game
    Let G be a finite group. On the first move, α chooses some g1 ∈ G. On the kth
    move, a player chooses gk
    ∈ G \ {g1, g2, . . . gk−1
    }. The loser is the player that
    generates all of G with their choice together with previous choices.
    Choice Set Generated
    g1 ⟨g1⟩
    g2 ⟨g1, g2⟩
    g3 ⟨g1, g2, g3⟩
    .
    .
    .
    .
    .
    .
    gk−1
    ⟨g1, g2, g3, . . . gk−1
    ⟩ ̸= G
    gk
    ⟨g1, g2, g3, . . . gk−1
    , gk
    ⟩ = G
    The player that chooses gk
    loses.
    D.C. Ernst Impartial games for generating groups 20 / 21
    

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21. DO NOT GENERATE Game for Arbitrary Groups
    Theorem (F.W. Barnes, 1988)
    Let G be any finite group with more than one element. Then α wins DO NOT
    GENERATE if and only if there is a g ∈ G such that
    • ⟨g⟩ has an odd number of elements;
    • ⟨g⟩ ̸= G;
    • ⟨g, h⟩ = G for all non-identity elements h ∈ G such that h2 = 1.
    Otherwise, β wins.
    Corollary
    • The player α wins DO NOT GENERATE on Zn
    if and only if n is odd or not a
    multiple of 4. Otherwise, β wins.
    • The player α wins DO NOT GENERATE on Dn
    if and only if n odd. Otherwise,
    β wins.
    D.C. Ernst Impartial games for generating groups 21 / 21
    

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