Dana Ernst
April 07, 2015
210

# Impartial achievement and avoidance games for generating finite groups

In this talk, we will explore two impartial games introduced by Anderson and Harary. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly selected elements wins the first game (achievement). The first player who cannot select an element without building a generating set loses the second game (avoidance). After the development of some general results, we determine the nim-numbers of both games for abelian and dihedral groups. In addition, we present a criteria on the maximal subgroups that determines the nim-numbers of avoidance games. Lastly, we apply our criteria to compute the nim-numbers of avoidance games for several families of groups, including nilpotent, generalized dihedral, generalized quaternion, and Coxeter groups. This is joint work with Bret Benesh and Nandor Sieben.

This talk was given on April 7, 2015 in the Northern Arizona University Department of Mathematics and Statistics Colloquium.

April 07, 2015

## Transcript

1. 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

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

3. 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

4. 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

5. our setup
∙ 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

6. nim
Single-pile Nim
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?
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

7. 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?
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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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