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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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