impartial achievement & avoidance games for generating finite groups NAU Department of Mathematics & Statistics Colloquium Dana C. Ernst Northern Arizona University April 7, 2015 Joint work with Bret Benesh and Nándor Sieben 

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. Note CGT should not be confused with the branch of economics called game theory. 1 

combinatorial game theory Combinatorial games ∙ Chess ∙ Go ∙ Connect Four ∙ Nim ∙ Tic-Tac-Toe ∙ X-Only Tic-Tac-Toe Non-combinatorial games ∙ Battleship (hidden information) ∙ Rock-Paper-Scissors (non-alternating and random) ∙ Poker (hidden information and random) 2 

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. Partizan ∙ Chess ∙ Go ∙ Connect Four ∙ Tic-Tac-Toe Impartial ∙ Nim ∙ X-Only Tic-Tac-Toe 3 

our setup Comments ∙ We are interested in impartial games. ∙ We will require that game sequence is finite and there are no ties. ∙ When analyzing games, we will assume that both players make optimal moves. ∙ Player that moves first is called α and second player is called β. ∙ Normal Play: The last player to move wins. ∙ Misère Play: The last player to move loses. 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. Game is denoted ∗n (called a nimber). . . . n Question Is there an optimal strategy for either player? Answer Boring: α always wins; just take the whole pile. Multi-pile Nim Start with k piles consisting of n1, . . . , nk stones, respectively. Each player chooses at least one stone from a single pile. The player that takes the last stone wins. Denoted ∗n1 + · · · + ∗nk . 5 

nim Example Let’s play ∗1 + ∗2 + ∗2. Here’s a possible sequence. (1, 2, 2) α → (0, 2, 2) β → (0, 1, 2) α → (0, 1, 1) β → (0, 1, 0) α → Yay! (0, 0, 0) In this case, α wins. Question Is there an optimal strategy for either player? Answer Short answer is yes: whittle down to an even number of piles with a single stone. If players make optimal moves, this is only possible for one of the players. 6 

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 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 If α plays in the middle square, the game is over quickly. 7 

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 misère 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. Check out “The Secrets of Notakto: Winning at X-only Tic-Tac-Toe” at http://arxiv.org/abs/1301.1672. ∙ Also, check out the free iPad app called Notakto. 8 

increasing rigor Definition An impartial game is a finite set X of positions together with a starting position and a collection {Opt(Q) ⊆ X | Q ∈ X} of possible options. Two players take turns choosing a single available option in Opt(Q) of current position Q. Player who encounters empty option set cannot move and loses. Note that positions are games in there own right. Definition Given a position, two possible states: ∙ P-position: previous player (player that just moved) wins ∙ N-position: next player (player that is about to move) wins From perspective of the player that is about to move, a P-position is a losing position while an N-position is a winning position. 9 

p-position vs n-position Examples ∙ ∗n is an N-position ∙ ∗1 + ∗1 is a P-position ∙ ∗1 + ∗2 is an N-position ∙ Empty X-Only Tic-Tac-Toe board is an N-position ∙ X-Only Tic-Tac-Toe board with X in middle is an P-position ∙ Two empty X-Only Tic-Tac-Toe boards is an P-position 10 

game sums Definition If G and H are games, then G + H is the game where each player makes a move in one of the games. Set of options: Opt(G + H) := {Q + H | Q ∈ Opt(G)} ∪ {G + S | S ∈ Opt(H)} Theorem G + G is a P-position. Proof Copy cat. 11 

game equivalence Definition G1 = G2 if G1 + G2 is a P-position. Intuition: something akin to “copy cat” works. Examples ∙ ∗1 + ∗1 = ∗0 since ∗1 + ∗1 + ∗0 is a P-position. ∙ ∗1 + ∗2 = ∗3 since ∗1 + ∗2 + ∗3 is a P-position. Theorem G1 = G2 iff G1 + H and G2 + H have the same outcome for all H. 12 

minimum excludant Definition If A is a set of ordinals, then mex(A) is the smallest ordinal not in A. Examples ∙ mex({0, 1, 2, 4, 5}) = 3 ∙ mex({1, 3}) = 0 ∙ mex({0, 1}) = 2 ∙ mex(∅) = 0 13 

nim-number of a game Definition If G is a game, then nim(G) := mex({nim(Q) | Q ∈ Opt(G)}). This is a recursive definition. We start computing with terminal positions (empty option set). Examples ∙ nim(∗0) = mex(∅) = 0 ∙ nim(∗1) = mex({nim(∗0)}) = mex({0}) = 1 ∙ nim(∗2) = mex({nim(∗0), nim(∗1)}) = mex({0, 1}) = 2 ∙ nim(∗n) = n ∙ nim(∗1 + ∗1) = mex({nim(∗1)}) = mex({1}) = 0 ∙ nim(∗1 + ∗2) = mex({nim(∗2), nim(∗1), nim(∗1 + ∗1)}) = mex({2, 1, 0}) = 3 14 

sprague–grundy theorem Theorem Every game is equivalent to a single Nim pile: G = ∗ nim(G). Examples ∙ ∗1 + ∗1 = ∗ nim(∗1 + ∗1) = ∗0 ∙ ∗1 + ∗2 = ∗ nim(∗1 + ∗2) = ∗3 Theorem β wins G iff G = ∗0. Big Picture Fundamental problem in the theory of impartial combinatorial games is the determination of the nim-number of the game. We can think of nim-numbers as “isomorphism” classes of games. 15 

achievement & avoidance games on finite groups Let G be a finite nontrivial group. Players take turns picking a new element gi ∈ G. Generate game (achievement) GEN(G): First player to build ⟨g1, g2, . . . , gi⟩ = G wins. Do Not Generate game (avoidance) DNG(G): The position ⟨g1, g2, . . . , gi⟩ = G is not allowed. Terminal positions are the maximal subgroups of G. Both games introduced by F. Harary in 1987. 16 

gen for cyclic groups Example GEN(Zn) is boring. Let’s look at GEN(Z6) with non-optimal play. Choice Set Generated a4 {a4, a2, e} 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 α can always win GEN(Zn) in one move by choosing any ak, where n and k are relatively prime. 17 

gen for dihedral groups Example Let’s look at GEN(D4). Choice Set Generated r2 {r2, e} e {r2, e} f {r2, e, f, r2f} r D4 β wins! 18 

dng for arbitrary groups Theorem (F.W. Barnes, 1988) Let G be any finite nontrivial group. Then α wins DNG(G) iff 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 = e. Otherwise, β wins. Corollary ∙ α wins DNG(Zn) iff n is odd or not a multiple of 4. Otherwise, β wins. ∙ α wins DNG(Dn) iff n odd. Otherwise, β wins. 19 

exploring nim-numbers ∅ ∗0 {0} ∗1 {2} ∗1 {0, 2} ∗0 ∅ ∗1 {1} ∗0 {0} ∗2 {2} ∗2 {3} ∗0 {0, 1} ∗0 {0, 2} ∗1 {0, 3} ∗0 {1, 2} ∗0 {2, 3} ∗0 {0, 1, 2} ∗0 {0, 2, 3} ∗0 DNG(Z4) = ∗0 GEN(Z4) = ∗1 20 

nim-numbers for cyclic groups Theorem (Ernst, Sieben) If n ≥ 2, then nim(GEN(Zn)) = nim(DNG(Zn)) + 1. Theorem (Ernst, Sieben) If n ≥ 2, then DNG(Zn) =              ∗1, n = 2 ∗1, n ≡2 1 ∗0, n ≡4 0 ∗3, n ≡4 2 and GEN(Zn) =              ∗2, n = 2 ∗2, n ≡2 1 ∗1, n ≡4 0 ∗4, n ≡4 2 21 

nim-numbers for dihedral groups Theorem (Ernst, Sieben) For n ≥ 3, we have DNG(Dn) = { ∗3, n ≡2 1 ∗0, n ≡2 0 and GEN(Dn) =        ∗3, n ≡2 1 ∗0, n ≡4 0 ∗1, n ≡4 2 22 

nim-numbers for abelian groups Theorem (Ernst, Sieben) If G is a finite nontrivial abelian group, then DNG(G) =              ∗1, G is nontrivial of odd order ∗1, G ∼ = Z2 ∗3, G ∼ = Z2 × Z2k+1 with k ≥ 1 ∗0, else GEN(G) =                              ∗2, |G| is odd and d(G) ≤ 2 ∗1, |G| is odd and d(G) ≥ 3 ∗2, G ∼ = Z2 ∗1, G ∼ = Z4k with k ≥ 1 ∗4, G ∼ = Z4k+2 with k ≥ 1 ∗1, G ∼ = Z2 × Z2 × Zm × Zk for m, k odd ∗0, else 23 

general results for dng Theorem (Ernst, Sieben) ∙ If G is any finite nontrivial group, then DNG(G) is ∗0, ∗1, or ∗3. ∙ If |G| is odd and nontrivial, then GEN(G) is ∗1 or ∗2. Conjecture If |G| is even, then GEN(G) is one of ∗0, ∗1, ∗2, ∗3, ∗4. 24 

general results for dng Theorem (Benesh, Ernst, Sieben) Let G be a finite nontrivial group. ∙ If |G| = 2, then DNG(G) = ∗1. ∙ If |G| is odd, then DNG(G) = ∗1. ∙ If |Φ(G)| is even, then DNG(G) = ∗0. ∙ If every maximal subgroup of G is even, then DNG(G) = ∗0. ∙ If the even maximal subgroups cover G, then DNG(G) = ∗0. ∙ Otherwise, DNG(G) = ∗3. Using our “checklist” criteria, we have completely characterized DNG for nilpotent, generalized dihedral, generalized quaternion, Coxeter (includes symmetric groups), alternating, and some Rubik’s cube groups. 25 

future work Open questions ∙ What is spectrum of GEN(G)? ∙ What is GEN(G) for generalized dihedral? (In progress) ∙ Are there nice results for products and quotients? (In progress) ∙ Is it possible to characterize the nim-numbers of GEN(G) in terms of covering conditions by maximal subgroups similar to what we did for DNG(G)? (In progress) Thanks! 26