Miserable MonoidsHow to Lose When You Must

Aaron N. Siegel & Thane E. Plambeck

aaron.n.siegel@gmail.com thane@best.com

Miserable Monoids – p. 1/22

Dawson’s Kayles (1)

Dawson’s Chess was invented by T. R. Dawson (1935).

An equivalent game is Dawson’s Kayles.

Interesting game so far.

But here Dawson made a huge mistake (depending on yourperspective).

Miserable Monoids – p. 2/22

Dawson’s Kayles (1)

Dawson’s Chess was invented by T. R. Dawson (1935).

An equivalent game is Dawson’s Kayles.

����@@@@

Interesting game so far.

But here Dawson made a huge mistake (depending on yourperspective).

Miserable Monoids – p. 2/22

Dawson’s Kayles (1)

Dawson’s Chess was invented by T. R. Dawson (1935).

An equivalent game is Dawson’s Kayles.

Interesting game so far.

But here Dawson made a huge mistake (depending on yourperspective).

Miserable Monoids – p. 2/22

Dawson’s Kayles (1)

Dawson’s Chess was invented by T. R. Dawson (1935).

An equivalent game is Dawson’s Kayles.

Interesting game so far.

But here Dawson made a huge mistake (depending on yourperspective).

Miserable Monoids – p. 2/22

Dawson’s Kayles (2)

In Dawson’s Kayles, whoever makes the last move loses.

This is the misère play condition.

Miserable Monoids – p. 3/22

Combinatorial Game Theory

We try to understand how games behave in combinations.

G = H iff G + X and H + X have the same outcomes,

for all games X that we might wish to consider in our theory.

For impartial games there are only two outcomes:

Previous player wins (P-positions)

Next player wins (N -positions)

Miserable Monoids – p. 4/22

Combinatorial Game Theory

We try to understand how games behave in combinations.

G = H iff G + X and H + X have the same outcomes,

for all games X that we might wish to consider in our theory.

For impartial games there are only two outcomes:

Previous player wins (P-positions)

Next player wins (N -positions)

Miserable Monoids – p. 4/22

Combinatorial Game Theory

We try to understand how games behave in combinations.

G = H iff G + X and H + X have the same outcomes,

for all games X that we might wish to consider in our theory.

For impartial games there are only two outcomes:

Previous player wins (P-positions)

Next player wins (N -positions)

Miserable Monoids – p. 4/22

The Sprague–Grundy Theory

Under the normal-play condition—last player wins—thetheory is extremely simple.

In the 1950s, R. P. Sprague and P. M. Grundy showed that

Every position in a normal-play impartial gameis equivalent to a nim-heap of size n, for some n.

Adding these Grundy values is a simple matter, and

G is a P-position iff its Grundy value is 0.

Miserable Monoids – p. 5/22

The Misère Theory (1)

. . . is vastly more complicated (Conway, 1970s).

We can make the same definition:

G = H iff G + X and H + X have the same outcomes,

for all misère games X.

But equivalences are rare.

Miserable Monoids – p. 6/22

The Misère Theory (1)

. . . is vastly more complicated (Conway, 1970s).

We can make the same definition:

G = H iff G + X and H + X have the same outcomes,

for all misère games X.

But equivalences are rare.

Miserable Monoids – p. 6/22

The Misère Theory (1)

. . . is vastly more complicated (Conway, 1970s).

We can make the same definition:

G = H iff G + X and H + X have the same outcomes,

for all misère games X.

But equivalences are rare.

Miserable Monoids – p. 6/22

The Misère Theory (2)

Let A be a set of games.

G ≡A H iff G + X and H + X have the same outcomes,

for all misère games X ∈ A .

A = the set of positions that arise in some game Γ (such asDawson’s Chess).

This often makes the theory manageable.

Conway, Sibert, and Allemang made some advances usingthis type of reduction.

Miserable Monoids – p. 7/22

The Misère Theory (2)

Let A be a set of games.

G ≡A H iff G + X and H + X have the same outcomes,

for all misère games X ∈ A .

A = the set of positions that arise in some game Γ (such asDawson’s Chess).

This often makes the theory manageable.

Conway, Sibert, and Allemang made some advances usingthis type of reduction.

Miserable Monoids – p. 7/22

The Misère Quotient of Γ

A = the set of positions in Γ.

If G ≡A H, then G + K ≡A H + K.

The set of equivalence classes has an additive structure!

Let Q = Q(A ) be this monoid.

Let P ⊂ Q correspond to P-positions.

The structure (Q,P) is the misère quotient of Γ.

Miserable Monoids – p. 8/22

The Misère Quotient of Γ

A = the set of positions in Γ.

If G ≡A H, then G + K ≡A H + K.

The set of equivalence classes has an additive structure!

Let Q = Q(A ) be this monoid.

Let P ⊂ Q correspond to P-positions.

The structure (Q,P) is the misère quotient of Γ.

Miserable Monoids – p. 8/22

A Simple Example

Γ = S(1, 2, 3) (misère subtraction game with set {1, 2, 3})

Q(Γ) ∼= {a, b : a2 = 1, b3 = b}; P = {a, b2}.

1 ∼ [G0, G4, G8, . . .]

a ∼ [G1, G5, G9, . . .]

b ∼ [G2, G6, G10, . . .]

ab ∼ [G3, G7, G11, . . .]

Miserable Monoids – p. 9/22

A Simple Example

Γ = S(1, 2, 3) (misère subtraction game with set {1, 2, 3})

Q(Γ) ∼= {a, b : a2 = 1, b3 = b}; P = {a, b2}.

1 ∼ [G0, G4, G8, . . .]

a ∼ [G1, G5, G9, . . .]

b ∼ [G2, G6, G10, . . .]

ab ∼ [G3, G7, G11, . . .]

Miserable Monoids – p. 9/22

Pretending Functions

The pretending function of Γ is the quotient mapΦ : A → Q(A ).

When Γ is a heap game we just give the single-heap valuesof Φ.

0 1 2 3

0 1 a b ab

4 1 a b ab

8 1 a · · ·

S(1,2,3)

Miserable Monoids – p. 10/22

Pretending Functions

The pretending function of Γ is the quotient mapΦ : A → Q(A ).

When Γ is a heap game we just give the single-heap valuesof Φ.

0 1 2 3

0 1 a b ab

4 1 a b ab

8 1 a · · ·

S(1,2,3)

Miserable Monoids – p. 10/22

Basic Results

Thane Plambeck showed that:

Q is often finite, even when A is infinite.

We can recover a winning strategy for Γ from (Q,P ,Φ).

Correctness of (Q,P ,Φ) can be verified algorithmically.

For octal games: if Φ is periodic for “long enough,” thenit’s periodic. (Misère analogue of the Guy–SmithPeriodicity Theorem.)

Nothing earth-shattering in the case of S(1, 2, 3), but . . .

We can handle lots of more complicated games that haveeluded researchers for decades.

Miserable Monoids – p. 11/22

Basic Results

Thane Plambeck showed that:

Q is often finite, even when A is infinite.

We can recover a winning strategy for Γ from (Q,P ,Φ).

Correctness of (Q,P ,Φ) can be verified algorithmically.

For octal games: if Φ is periodic for “long enough,” thenit’s periodic. (Misère analogue of the Guy–SmithPeriodicity Theorem.)

Nothing earth-shattering in the case of S(1, 2, 3), but . . .

We can handle lots of more complicated games that haveeluded researchers for decades.

Miserable Monoids – p. 11/22

Basic Results

Thane Plambeck showed that:

Q is often finite, even when A is infinite.

We can recover a winning strategy for Γ from (Q,P ,Φ).

Correctness of (Q,P ,Φ) can be verified algorithmically.

For octal games: if Φ is periodic for “long enough,” thenit’s periodic. (Misère analogue of the Guy–SmithPeriodicity Theorem.)

Nothing earth-shattering in the case of S(1, 2, 3), but . . .

We can handle lots of more complicated games that haveeluded researchers for decades.

Miserable Monoids – p. 11/22

Basic Results

Thane Plambeck showed that:

Q is often finite, even when A is infinite.

We can recover a winning strategy for Γ from (Q,P ,Φ).

Correctness of (Q,P ,Φ) can be verified algorithmically.

For octal games: if Φ is periodic for “long enough,” thenit’s periodic. (Misère analogue of the Guy–SmithPeriodicity Theorem.)

Nothing earth-shattering in the case of S(1, 2, 3), but . . .

We can handle lots of more complicated games that haveeluded researchers for decades.

Miserable Monoids – p. 11/22

Basic Results

Thane Plambeck showed that:

Q is often finite, even when A is infinite.

We can recover a winning strategy for Γ from (Q,P ,Φ).

Correctness of (Q,P ,Φ) can be verified algorithmically.

For octal games: if Φ is periodic for “long enough,” thenit’s periodic. (Misère analogue of the Guy–SmithPeriodicity Theorem.)

Nothing earth-shattering in the case of S(1, 2, 3), but . . .

We can handle lots of more complicated games that haveeluded researchers for decades.

Miserable Monoids – p. 11/22

MisereSolver

Software to calculate misère quotients of octal games.

Misère Guiles (0.15) was unsolved until recently.MisereSolver can solve it in under 5 seconds.

Miserable Monoids – p. 12/22

Guiles (1)

Q ∼= {a, b, c, d, e, f, g, h, i | a2 = 1, b4 = b2, bc = ab3, c2 = b2,

b2d = d, cd = ad, d3 = ad2, b2e = b3, de = bd, be2 = ace,

ce2 = abe, e4 = e2, bf = b3, df = d, ef = ace, cf2 = cf,

f3 = f2, b2g = b3, cg = ab3, dg = bd, eg = be, fg = b3,

g2 = bg, bh = bg, ch = ab3, dh = bd, eh = bg, fh = b3,

gh = bg, h2 = b2, bi = bg, ci = ab3, di = bd, ei = be, fi = b3,

gi = bg, hi = b2, i2 = b2}

P = {a, b2, bd, d2, ae, ae2, ae3, af, af2, ag, ah, ai}

Miserable Monoids – p. 13/22

Guiles (2)

0 1 2 3 4 5 6 7 8 9

0 1 a a 1 a a b b a b

10 b a a 1 c c b b d b

20 e c c f c c b g d h

30 i ab2 abg f abg abe b3 h d h

40 h ab2 abe f2 abg abg b3 h d h

50 h ab2 abg f2 abg abg b3 b3 d b3

60 b3 ab2 abg f2 abg abg b3 b3 d b3

70 b3 ab2 ab2 f2 ab2 ab2 b3 b3 d b3

80 b3 ab2 ab2 f2 ab2 ab2 b3 b3 d b3

90 b3 ab2 ab2 f2 · · ·

Miserable Monoids – p. 14/22

Nontrivial (“Wild”) Three-Digit Octals

Game Period Ppd |Q|

0.15 10 66 420.34 8 7 120.53 9 21 160.71 6 3 360.72 4 16 240.75 2 8 80.77 12 71 400.115 14 92 420.123 5 5 20

Game Period Ppd |Q|

0.152 48 25 340.153 14 32 160.241 10 4 360.351 8 4 220.512 6 16 80.712 6 3 140.716 2 22 144.56 2 11 8

Miserable Monoids – p. 15/22

0.4107 (1)

The game 0.4107:

Q has 506 elements (58 in P)

A minimal set of 34 generators

Nonetheless, it settles down rather quickly!

Miserable Monoids – p. 16/22

0.4107 (2)

0 1 2 3 4 5 6 7 8 9 10 11

0 1 1 a a b b ab c c d e f

12 g h b i ab2 j k l m n o p

24 q r abo anq b3 s t abm cq2 u cjk v

36 w x b3 y agt z b2i A B b3 C D

48 b4c bco abF E b3 F ab3c grx G abF abF b3

60 ab3c H b3 b4c ab4 cfH b4c ab3c ab3c b3 b3 ab4

72 b4c b4c ab4 ab3c b3 b3 ab3c b4c b4c ab4 ab4 b3

84 ab3c ab3c b3 b4c ab4 ab4 b4c ab3c ab3c b3 b3 ab4

96 b4c b4c ab4 ab3c b3 b3 ab3c b4c b4c ab4 ab4 b3

108 ab3c ab3c b3 b4c ab4 ab4 b4c ab3c ab3c b3 b3 ab4

120 b4c b4c ab4 ab3c b3 b3 ab3c b4c b4c ab4 ab4 b3

132 ab3c ab3c b3 b4c ab4 ab4 b4c ab3c ab3c b3 · · ·

Miserable Monoids – p. 17/22

Dawson’s Kayles (3)

“Easy” to heap 24. Thereafter the quotient size growsrapidly.

Max Heap |Q|

24 2426 14429 17630 36031 52032 55233 63834 ∞?

Miserable Monoids – p. 18/22

Q in General

Q(A ) can be infinite even when A is finitely generated

If Q is finite and non-trivial, it has even order

Conjecture: Every element has period 1, 2 or ∞(The period of x is the least n for which xk+n = xk, forsome k)

Miserable Monoids – p. 19/22

Q in General

Q(A ) can be infinite even when A is finitely generated

If Q is finite and non-trivial, it has even order

Conjecture: Every element has period 1, 2 or ∞(The period of x is the least n for which xk+n = xk, forsome k)

Miserable Monoids – p. 19/22

Q in General

Q(A ) can be infinite even when A is finitely generated

If Q is finite and non-trivial, it has even order

Conjecture: Every element has period 1, 2 or ∞(The period of x is the least n for which xk+n = xk, forsome k)

Miserable Monoids – p. 19/22

Classification Theory

How many (non-isomorphic) misère quotients of order 2n?

Just one of order 2: Z2

None of order 4

One of order 6: T2 (easy to show)

One of order 8 (much harder)

Conjecture: one of order 10; four of order 12

Miserable Monoids – p. 20/22

Classification Theory

How many (non-isomorphic) misère quotients of order 2n?

Just one of order 2: Z2

None of order 4

One of order 6: T2 (easy to show)

One of order 8 (much harder)

Conjecture: one of order 10; four of order 12

Miserable Monoids – p. 20/22

Classification Theory

How many (non-isomorphic) misère quotients of order 2n?

Just one of order 2: Z2

None of order 4

One of order 6: T2 (easy to show)

One of order 8 (much harder)

Conjecture: one of order 10; four of order 12

Miserable Monoids – p. 20/22

Classification Theory

How many (non-isomorphic) misère quotients of order 2n?

Just one of order 2: Z2

None of order 4

One of order 6: T2 (easy to show)

One of order 8 (much harder)

Conjecture: one of order 10; four of order 12

Miserable Monoids – p. 20/22

Classification Theory

How many (non-isomorphic) misère quotients of order 2n?

Just one of order 2: Z2

None of order 4

One of order 6: T2 (easy to show)

One of order 8 (much harder)

Conjecture: one of order 10; four of order 12

Miserable Monoids – p. 20/22

The Seven Misère Quotients Born by Day 4

Quotient # Presentation P-partition Gen

Z2 2 {a | a2 = 1} {a} 1

T2 6 {a, b | a2= 1, b3 = b} {a, b2} 2

R8 8 {a, b, c | a2 = 1, b3 = b, bc = b, c2 = b2} {a, b2} 2+320

T3 10 {a, b, c | a2= 1, b3 = b, c3 = c, c2 = b2} {a, b2} 4

T2 × Z2 12 {a, b, c | a2 = 1, b3 = b, c2 = 1} {a, b2, ac} 2+321

S12 12 {a, b, c | a2= 1, b4 = b2, b2c = b3, c2 = 1} {a, b2, ac} 2+1

R14 14 {a, b, c | a2 = 1, b3 = b, b2c = c, c3 = c2} {a, b2, bc, c2} 2+0

Miserable Monoids – p. 21/22

Three Major Open Problems

Give a computational method to determine whether aquotient is infinite.

Give a method for computing presentations of infinitequotients. This appears to be necessary if we hope tosolve Dawson’s Kayles.

Some octal games are not periodic; their partialquotients, though finite, grow ever larger. Nonetheless,their pretensions exhibit strong algebraic regularity.Formulate and prove an “algebraic periodicity” theoremfor such quotients.

Miserable Monoids – p. 22/22

Three Major Open Problems

Give a computational method to determine whether aquotient is infinite.

Give a method for computing presentations of infinitequotients. This appears to be necessary if we hope tosolve Dawson’s Kayles.

Some octal games are not periodic; their partialquotients, though finite, grow ever larger. Nonetheless,their pretensions exhibit strong algebraic regularity.Formulate and prove an “algebraic periodicity” theoremfor such quotients.

Miserable Monoids – p. 22/22

Three Major Open Problems

Give a computational method to determine whether aquotient is infinite.

Give a method for computing presentations of infinitequotients. This appears to be necessary if we hope tosolve Dawson’s Kayles.

Some octal games are not periodic; their partialquotients, though finite, grow ever larger. Nonetheless,their pretensions exhibit strong algebraic regularity.Formulate and prove an “algebraic periodicity” theoremfor such quotients.

Miserable Monoids – p. 22/22