INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5 (2005), #G01

SUMBERS – SUMS OF UPS AND DOWNS

Kuo-Yuan Kao

National Penghu Institute of Technology, Taiwan

                            stone@npit.edu.tw

Received: 4/23/04, Revised: 1/20/05, Accepted: 03/7/05, Published: 3/8/05

Abstract

A sumber is a sum of ups, downs and star. We provide a simplification rule that can determine whether a game G is a sumber or not, and if it is, determine the exact sumber of G from its left and right options, G^{L} and G^{R}. Once one knows a game is a sumber, the outcome of the game can be determined by a simple rule.

1. Introduction

We are concerned with combinatorial games and follow the notations and conventions of Winning Ways [1]. A game G is an ordered pair of sets of games,

G = {G^{L}|G^{R}},

where G^{L} and G^{R} are called the left and right options of G. We assume the readers are familiar with the birthdays [2] of games, and say a game is simpler if it was born earlier. The simplest game is the game 0, defined as

0 = { | }.

The following are the basic definitions of combinatorial games:

−G = {−G^{R}| − G^{L}}.
G ≥ H iff no element in G^{R} ≤ H.
G ≤ H iff no element in G^{L} ≥ H.
G = H iff G ≥ H and G ≤ H.
G + H = {G^{L} + H, G + H^{L}|G^{R} + H, G + H^{R}}.

These few lines already define a group with a very rich structure. The properties of zero, negation, order, equality and addition, can be proven to have the usual [1][2] behaviors.

There is a natural interpretation of the order defined above. Consider a game G played by two players, L and R, who move alternatively. The player who makes the last legal move is the winner. The outcome of the game can be describe as:

G ≥ 0, if R cannot win the game playing first,

G ≤ 0, if L cannot win the game playing first,

G = 0, if the first player cannot win the game, and

G ≱ H and G ≰ H, if the first player can win the game.

To ease the discussion, we include the following definitions.

G || H iff G ≱ H and G ≰ H,
G > H iff G ≥ H and G ≠ H,
G < H iff G ≤ H and G ≠ H,
G | > H iff G ≰ H,
G < | H iff G ≱ H.

For any game G, the outcome is in one of the four cases: G > 0, G < 0, G = 0, or G || 0.

In general, it is not an easy task to determine the outcome of a complex game. We are interested in subgroups of games with the following properties:

1. There exists a simple rule to determine the outcome of any game in the subgroup.

2. There exists a simplification rule that can simplify games in the subgroup.

For examples, numbers [2] and nimbers [3] are well known subgroups with the above properties. The simplest number rule [2] and the minimal excluded rule [4] can simplify numbers and nimbers respectively.

In this article, we study another subgroup of games called sumbers. Sumbers also have the properties mentioned above. The rest of this article is organized as follows. In section 2, we introduce the constituting elements of a sumber : ups, downs and star. We define the minimum cut of a sumber and the critical section of a game whose options are sumbers. The critical section plays an important role in determining whether a game is a sumber or not. In section 3, we present the major result for single-option games, together with the proofs. In section 4, the result is extended to cover multi-option games. Examples are provided in section 5.

2. Sumbers

2.1 Ups and Downs

For any number d, define

↑ (d) = {↑ (d^{L}), ∗|∗, ↑ (d^{R})}

where ∗ = {0|0} (pronounced star). When d = 0, we have

↑ (0) = {∗|∗} = 0.

By simple induction, one can show that, for any number d,

↑ (−d) = {↑ (−d^{R}), ∗|∗, ↑ (−d^{L})} = {− ↑ (d^{R}), ∗|∗, − ↑ (d^{L})} = − ↑ (d).

Each ↑ (d), d > 0, is called an up and has atomic weight 1 [2] (chapter 16, page 218). The negation of an up is called a down. The number d is called the order of ↑ (d).

We use the notation n. ↑ (d) to denote the sum of n copies of ↑ (d):

0. ↑ (d) = 0
n. ↑ (d) = (n − 1). ↑ (d)+ ↑ (d), n > 0.

The ups defined above have the following properties:

Proposition 1: ↑ (d_{2}) >↑ (d_{1}), for all d_{2} > d_{1}.

Proof: We prove by induction. When d_{1} = 0, the claim is clearly true. Otherwise, suppose the claim is true for simpler games. Since d_{2} > d_{1}, we have d_{2} > d_{1}^{L} and d_{2}^{R} > d_{1}. By induction hypothesis, we have ↑ (d_{2}) >↑ (d_{1}^{L}) and ↑ (d_{2}^{R}) >↑ (d_{1}). This implies R cannot win the game ↑ (d_{2})− ↑ (d_{1}) playing first. Thus ↑ (d_{2}) >↑ (d_{1}). □

Proposition 2: ↑ (d_{2})+ ↑ (d_{2})− ↑ (d_{1}) + ∗ > 0, for all d_{2} > d_{1} > 0.

Proof: Consider the game G =↑ (d_{2})+ ↑ (d_{2})− ↑ (d_{1}) + ∗. L can win G by moving to ↑ (d_{2})+ ↑ (d_{2}). R cannot win G by moving to ↑ (d_{2}^{R})+ ↑ (d_{2})− ↑ (d_{1}) + ∗, ↑ (d_{2})− ↑ (d_{1}), ↑ (d_{2})+ ↑ (d_{2})− ↑ (d_{1}^{L}) + ∗, ↑ (d_{2})+ ↑ (d_{2}), ↑ (d_{2})+ ↑ (d_{2})− ↑ (d_{1}). Thus ↑ (d_{2})+ ↑ (d_{2})− ↑ (d_{1}) + ∗ > 0. □

Proposition 3: ↑ (d_{2})− ↑ (d_{1}) > n.(↑ (d_{3})− ↑ (d_{2})), for all d_{3} > d_{2} > d_{1} > 0 and n ≥ 0.

Proof: We prove by induction on n. When n = 0, the claim is clearly true. Otherwise, suppose the claim is true when n = k. Let d be a number such that d_{2} > d > d_{1}. By induction hypothesis, we have ↑ (d_{2})− ↑ (d) > k.(↑ (d_{3})− ↑ (d_{2})) and ↑ (d)− ↑ (d_{1}) >↑ (d_{2})− ↑ (d) >↑ (d_{3})− ↑ (d_{2}). Thus ↑ (d_{2})− ↑ (d_{1}) > (k + 1).(↑ (d_{3})− ↑ (d_{2})). □

A sum of ups, downs and star is called a sumber. A sumber S can be written in the standard form:

S = s_{0}. ∗ + ∑_{k=1,n} s_{k}. ↑ (d_{k}),

where s_{0} = 0 or 1, s_{k} ≠ 0, 0 < k ≤ n and 0 < d_{k} < d_{k+1}, 0 < k < n. ∑_{k=1,n} s_{k} is called the net weight of S.

Sumbers are closed under addition. Conway first found this interesting subgroup of games. He also found a rule [2](theorem 88, page 194) similar to the theorem below to determine the outcome of a sumber.

Theorem 1: Sumber outcome rule. Let S be a sumber in the above standard form.

• S > 0 if and only if (∑_{k=1,n} s_{k} > s_{0}) or (∑_{k=1,n} s_{k} = s_{0} and s_{1} < 0).

• S < 0 if and only if −S > 0

• S = 0 if and only if n = 0 and s_{0} = 0.

• S | 0 , otherwise.

Proof: We prove the if part of the first case. The rest is trival once the first case been proven. If ∑_{k=1,n} s_{k} > s_{0}, let d be a number such that d_{1} > d > 0, then

S = s_{0}. ∗ + ∑_{k=1,n} s_{k}. ↑ (d_{k})

> s_{0}. ∗ + ∑_{k=1,n} |s_{k}|.(↑ (d_{1})− ↑ (d_{n})) + ∑_{k=1,n} s_{k}. ↑ (d_{1})

> s_{0}. ∗ + ↑ (d)− ↑ (d_{1}) + (s_{0} + 1). ↑ (d_{1})
= s_{0}. ∗ + ↑ (d) + s_{0}. ↑ (d_{1}) > 0.

If ∑_{k=1,n} s_{k} = s_{0} and s_{1} < 0, let d be a number such that d_{2} > d > d_{1}, then

S = s_{0}. ∗ + ∑_{k=1,n} s_{k}. ↑ (d_{k})

> s_{0}. ∗ + ∑_{k=2,n} |s_{k}|.(↑ (d_{2})− ↑ (d_{n})) + ∑_{k=2,n} s_{k}. ↑ (d_{2}) + s_{1}. ↑ (d_{1})

> s_{0}. ∗ + ↑ (d)− ↑ (d_{2}) + (s_{0} − s_{1}). ↑ (d_{2}) + s_{1}. ↑ (d_{1})
> s_{0}. ∗ + ↑ (d) + s_{0}. ↑ (d_{2})− ↑ (d_{1}) > 0.
□

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5 (2005), #G01 5

2.2. Minimum Cut and Critical Section

Definition 1: Cut of a sumber. Let S be a sumber written in the standard form:

S = s_{0}. ∗ + ∑_{k=1,n} s_{k}. ↑ (d_{k}),

where s_{0} = 0 or 1, s_{k} ≠ 0, 0 < k ≤ n and 0 < d_{k} < d_{k+1}, 0 < k < n. For m ∈ {0, d_{k} : 1 ≤ k ≤ n}, define

S^{m} = ∑_{k=1,n; d_{k}≥m} s_{k}. ↑ (d_{k}) − ∑_{k=1,n; d_{k}≥m} s_{k}. ↑ (m).

We say S has a cut at m if S^{m} ≤ 0. Example 1: Consider the sumber S = 4. ↑ (1) − 2. ↑ (2)+ ↑ (3)+ ↑ (5)− ↑ (6). S^{0} = 4. ↑ (1) − 2. ↑ (2)+ ↑ (3)+ ↑ (5)− ↑ (6) − 3. ↑ (0) > 0, S^{1} = 4. ↑ (1) − 2. ↑ (2)+ ↑ (3)+ ↑ (5)− ↑ (6) − 3. ↑ (1) < 0,

S^{2} =  −2. ↑ (2)+ ↑ (3)+ ↑ (5)− ↑ (6)  + 1. ↑ (2) > 0,
S^{3} =  ↑ (3)+ ↑ (5)− ↑ (6)  − 1. ↑ (3) < 0,
S^{5} =  ↑ (5)− ↑ (6)  + 0. ↑ (5) < 0,
S^{6} =  − ↑ (6)  + 1. ↑ (6) = 0.

Thus, S has cuts at 1, 3, 5 and 6. □

We only concerned with the minimum cut: the smallest number in {0, d_{k} : 1 ≤ k ≤ n} which is a cut. When S is a sumber, and m is the minimum cut of S, we call S^{m} the upper section and S_{m} = S − S^{m} the lower section of S. Example 2: Consider the sumber S = 4. ↑ (1) − 2. ↑ (2)+ ↑ (3)+ ↑ (5)− ↑ (6). From example 1, we know S has cuts at 1, 3, 5 and 6. The minimum cut of S is at 1. The upper section of S is S^{1} =↑ (1) − 2. ↑ (2)+ ↑ (3)+ ↑ (5)− ↑ (6), and the lower section is S_{1} = 3. ↑ (1). □

Definition 2: Critical section of {A|B}. Let A, B be two sumbers, define X(A|B) = {x ≥ m : A − B_{m} + ∗ < | ↑ (x) < | B^{m} + ∗, m is the minimum cut of B}. X(A|B) is called the critical section of the game {A|B}. The calculation of X(A|B) involves solving inequalities of sumbers. Theorem 1 can help in solving these inequalities.

Example 3: Consider the game {0|S} = {0 | 4. ↑ (1) − 2. ↑ (2)+ ↑ (3)+ ↑ (5)− ↑ (6)}.

From example 2, we know the minimum cut of S is at 1.

X(0|S) = {x ≥ 1 : 0 − S_{1} + ∗ < | ↑ (x) < | S^{1} + ∗}

= {x ≥ 1 : −3. ↑ (1) + ∗ < | ↑ (x) < | ↑ (1) − 2. ↑ (2)+ ↑ (3)+ ↑ (5)− ↑ (6) + ∗}

= {1}.
□

One of the interesting results about ups is the following up-star equality [2]:

{0| ↑ (1)} =↑ (1)+ ↑ (1) + ∗.

The interesting point is that certain games can be decomposed as sums of ups, downs and star.

Kao [5] also designed a game played on a number of heaps of colored counters. Each counter is colored either black or white. Left and Right move alternatively and their legal moves are different:

• When it is L’s turn to move, he can choose any one of the heaps and repeatly removes the top counter of that heap until either he removed a white counter or the heap has become empty.

• When it is R’s turn to move, he can choose any one of the heaps and repeatly removes the top counter of that heap until either he removed a black counter or the heap has become empty.

The player who remove the last counter is the winner. Kao [5] found that each of the colored heaps can be decomposed as a sum of ups, downs and star.

These exciting results raise the questions: what is the general rule that a game can be decomposed as a sum of ups, downs and star? Can we obtain a result similar to the simplest number rule for numbers or the minimal excluded value rule for nimbers? As we shall see in the next section, the answer is yes and X(A|B) plays an important role in determining whether {A|B} is a sumber or not.

3. Sumber Simplification Rule

If A and B are sumbers then the following rule can determine whether {A|B} is a sumber or not.

Theorem 2: Sumber simplification rule (single-option games).

Let A, B be two sumbers.

1. If A < | 0 and B | > 0 then {A|B} = 0.

2. Otherwise, either A ≥ 0 or B ≤ 0. Without loss of generality, we may assume A ≥ 0 and the net weight of A + B is greater than or equal to 0 (otherise, apply this rule to −{A|B}). By assumption, the net weight of B is greater than or equal to −1 (otherise, {A|B} is not a sumber).

(a) If B has non-negative net weight then {A|B} is a sumber if and only if X(A|B) is not empty.

(b) If B has net weight −1 and B || 0 then {A|B} is a sumber if and only if X(A|B) is not empty.

(c) Otherwise, B has net weight −1, and B < 0. If A||∗ and B||∗, then {A|B} = ∗.

(d) Otherwise, B has net weight −1, either A > ∗ or B < ∗. Without loss of generality, we may assume A > ∗ (otherise, apply this rule to −{A|B}). {A|B} is a sumber if and only if X(A|B) is not empty.

In cases 2(a), 2(b), and 2(d), If {A|B} is a sumber then {A|B} = B_{m} + ∗+ ↑ (p) where m is the minimum cut of B and p is the simplest number in X(A|B).

The proofs for the above cases are provided in propositions 4, 5, and 6.

Lemma 1: If B is a sumber with non-negative net weight and m ≥ 0 is the minimum cut of B, then, whenever L removes a negative term from B_{m}, R can find a higher or equal order positive term from the remaining sum to remove.

Proof: When m = 0, since B has non-negative net weight, we have either B_{m} = 0 or B_{m} = ∗. In either case, there is no negative term in B_{m} and the claim is obviously true.

When m > 0, we prove the claim by contradiction. Consider the game B_{m}. Suppose, after some pair(s) of moves, L chooses − ↑ (b) in the remaining sum and the most positive term left is ↑ (a), 0 < a < b < m. By assumption, B_{m} must be of the form:

B_{m} = (terms with order ≤ a) + ↑ (a)− ↑ (b)+ (terms with order ≥ b, net weight = 0).

But this implies a < m is a cut of B and m is not the minimum cut, a contradiction. □

Lemma 2: If B is a sumber with non-negative net weight, and m > 0 is the minimum cut of B, then {0|B} − B_{m}+ ↑ (m)− ↑ (n) ≥ 0, for all n < m.

Proof: It is sufficient to show that L cannot win the sum S = {−B| 0} + B_{m}− ↑ (m)+ ↑ (n). Without loss of generality, we may assume n is a number greater than the order of the most negative term in B_{m}.

1. L cannot win S by choosing the {−B| 0} option, because −B + B_{m}− ↑ (m)+ ↑ (n) = −B^{m}− ↑ (m)+ ↑ (n) < 0 (the sum has net weight 0, without ∗, and the smallest term, ↑ (n), is positive,).

2. L cannot win S by choosing any negative term from B_{m}− ↑ (m)+ ↑ (n), because, by lemma 1, whenever L removes a negative term from B_{m}− ↑ (m)+ ↑ (n), R can find a higher or equal order positive term from the remaining sum to remove. (B_{m}− ↑ (m)+ ↑ (n) has non-negative net weight and it equals its lower section.)

3. L cannot win S by choosing any positive term from B_{m}− ↑ (m)+ ↑ (n). From atomic weight calculus [1], we know the atomic weight of {0|B} eqauls −1, 0 or [1 + the atomic weight of B], depending on {0|B} is less than, confused with, or greater than the remote star. Since B has non-negative atomic weight and m > 0, we know B ≰ 0, and B ≰ ∗, thus {0|B} > 0 and {0|B} > ∗, the atomic weight of {0|B} must equals [1 + the atomic weight of B]. This implies the atomic weight of S is −1. So, L cannot win S by choosing some positive term from B_{m}− ↑ (m)+ ↑ (n).

From 1, 2 and 3, we know L cannot win S. □

Lemma 3: If A ≥ 0 is a sumber, B is a sumber with non-negative net weight, m ≥ 0 is the minimum cut of B, then {A|B} − B_{m} + ∗− ↑ (n) > 0, for all n < m.

Proof: Consider the sum S = {A|B} − B_{m} + ∗− ↑ (n).

When m = 0, since B has non-negative net weight, we have either B_{m} = 0 or B_{m} = ∗. Note that in this case n < 0. If B_{m} = 0 then S = {A|B^{m}} + ∗− ↑ (n) > 0. If B_{m} = ∗ then S = {A|B^{m} + ∗}− ↑ (n) > 0 . In either case, we have S > 0.

When m > 0, we need to show that L can and R cannot win S. Without loss of generality, we may assume n is a number greater than the order of the most negative term in B_{m}. L can win S by choosing the − ↑ (m) term in −B_{m}, because, after L’s move, the sum becomes {A|B} − B_{m}+ ↑ (m)− ↑ (n) ≥ {0|B} − B_{m}+ ↑ (m)− ↑ (n) ≥ 0 (by lemma 2). R cannot win S by choosing the {A|B} option, because, after R’s move, the sum becomes B − B_{m} + ∗− ↑ (n)|| 0 (there exists ∗, net weight = −1, and the least order term is negative). R cannot win S by choosing any positive term from −B_{m}, because, by lemma 1, whenever R removes a positive term from −B_{m}, L can find a higher order negative term in the remaining sum to remove. Thus, {A|B} − B_{m} + ∗− ↑ (n) > 0. □

Proposition 4: If A ≥ 0 is a sumber, B is a sumber with non-negative net weight and X(A|B) is not empty, then {A|B} is a sumber. Moreover {A|B} = B_{m} + ∗+ ↑ (p) where m is the minimum cut of B and p is the simplest number in X(A|B).

Proof: To prove {A|B} = B_{m} + ∗+ ↑ (p), it is sufficient to show that the first player cannot win the game G = {A|B} − B_{m} + ∗− ↑ (p).

Claim 1: L cannot win the game G.

L can choose terms from {A|B}, − ↑ (p), −B_{m} or ∗. Since the later two are dominated options, we only need to consider the first two terms.

1. L cannot win G by choosing {A|B}, because, after L’s move, the remaining game will be A − B_{m} + ∗− ↑ (p) < | 0 (because p ∈ X(A|B)).

2. L cannot win G by choosing − ↑ (p), because, after L’s move, the remaining game will be {A|B} − B_{m} < | 0 (because B ≤ B_{m}).

In either case, L cannot win the game G.

Claim 2: R cannot win the game G.

R can choose terms from {A|B}, − ↑ (p), −B_{m} or ∗.

1. R cannot win G by choosing {A|B}, because, after R’s move, the remaining game will be B − B_{m} + ∗− ↑ (p)| > 0 (because p ∈ X(A|B)).

2. If R chooses the − ↑ (p) term then the remaining game will be {A|B}−B_{m} +∗− ↑ (p^{L}). Since p is the simplest number in X(A|B), we have either p^{L} ∈ X(A|B) or p^{L} ∉ X(A|B).

(a) If p^{L} ∉ X(A|B) then A − B_{m} + ∗− ↑ (p^{L}) ≥ 0, thus {A|B} − B_{m} + ∗− ↑ (p^{L})| > 0.

(b) If p^{L} ∈ X(A|B) then p = m, thus {A|B} − B_{m} + ∗− ↑ (p^{L}) = {A|B} − B_{m} + ∗− ↑ (m^{L}) > 0 (by lemma 3).

3. If R chooses a positive term, say ↑ (n), from −B_{m} then L can respond a move at − ↑ (p). After the exchange, the remaining game becomes {A|B} − B_{m} + ∗− ↑ (n) > 0 (by lemma 3). R should never consider choosing any negative terms from −B_{m} because these options are dominated by the − ↑ (p) option.

4. If R chooses the ∗ term then L can respond a move at − ↑ (p). After the exchange, the remaining game becomes {A|B} − B_{m} + ∗ > 0.

In any of the 4 cases, R cannot win the game G. □

Proposition 5: If A ≥ 0, B || 0 are sumbers, B has net weight −1 and X(A|B) is not empty, then {A|B} = B_{m} + ∗+ ↑ (p), where m is the minimum cut of B and p is the simplest number in X(A|B).

Proof: Since B || 0 and B’s net weight is −1, we know the least order term in B is negative. Let − ↑ (q) be the least order term in B. Now, consider the sum S = {A+ ↑ (q)|B+ ↑ (q)}− ↑ (q) + {−B| − A}. We want to show that the first player cannot win S.

By proposition 4, we know {A+ ↑ (q)|B+ ↑ (q)} = B_{m}+ ↑ (q) + ∗+ ↑ (p), where m is the minimum cut of B+ ↑ (q) and p is the simplest number in X(A+ ↑ (q)|B+ ↑ (q)). Note that the minimum cut of B+ ↑ (q) equals the minimum cut of B and X(A+ ↑ (q)|B+ ↑ (q)) = X(A|B). We can rewrite S as S = B_{m}+ ↑ (q) + ∗+ ↑ (p)− ↑ (q) + {−B| − A}. Thus, − ↑ (q) is a dominated option for both players. If a player can win S, then the player must have a winning move other than − ↑ (q). Since neither of the players can win S by moving to {A+ ↑ (q)|B+ ↑ (q)} or {−B| − A}, S = 0. Also note that the minimum cut if B equals the minimum cut of B+ ↑ (q). We have {A|B} = B_{m} + ∗+ ↑ (p), where m is the minimum cut of B and p is the simplest number in X(A|B). □

Proposition 6: If A > 0 is a sumber with net weight 1, B < 0 is a sumber with net weight −1, and X(A|B) is not empty, then {A|B} is a sumber. Moreover, if A > ∗ then {A|B} = B_{m} + ∗+ ↑ (p), where m is the minimum cut of B and p is the simplest number in X(A|B).

Proof: Since X(A|B) is not empty, we have either A || ∗ or B || ∗ (If A > ∗ and B < ∗ then X(A|B) is empty.) If A || ∗ and B || ∗ then {A|B} = ∗. Otherwise, either (A > ∗ and B||∗) or (A||∗ and B < ∗). Without loss of generality, we may assume A > ∗ and B||∗. Consider the sum S = {A + ∗|B + ∗} + ∗ + {−B| − A}. We want to show that the first player cannot win S. Since A + ∗ > 0, B + ∗ has net weight −1 and B + ∗ || 0, by proposition 5, we have {A + ∗|B + ∗} = B_{m}+ ↑ (p) where m is the minimum cut of B and p is the simplest number in X(A|B). We can rewrite S as S = B_{m}+ ↑ (p) + ∗ + {−B| − A}. Note that ∗ is a dominated option for both players. If a player can win S, then the player must have a winning move other than ∗. Since neither of the players can win S by moving to {A + ∗|B + ∗} or {−B| − A}, we have S = 0. Thus, {A|B} = B_{m} + ∗+ ↑ (p), where m is the minimum cut of B and p is the simplest number in X(A|B). □

4. Games with Multiple Options

Section 3 deals with games where each player has a unique option. In this section, we extend the theorem to deal with multiple-option games. Note that, sums of ups and downs (excluding *) are totally ordered. If G^{L} and G^{R} are sets of sumbers, then G has at most two non-dominated options, one contains ∗ and the other does not, in each of G^{L} and G^{R}. In other words, G can be simplified as G = {A, B|C, D} where A, B are the non-dominated options in G^{L} and C, D are the non-dominated options in G^{R}. Without loss of generality, we may assume the net weight of C is less than or equal to the net weight of D.

The critical section X(G) of G = {A, B|C, D} is defined as empty set, if C and D has the same net weight. Otherwise (the net weight of C is less than the net weight of D), X(G) is defined as the set of numbers x ≥ m satisfying all the following inequalities:

↑ (x) | > A − C_{m} + ∗,
↑ (x) | > B − C_{m} + ∗,
↑ (x) < | C − C_{m} + ∗,
↑ (x) < | D − C_{m} + ∗,

where m is the minimum cut of C.

Theorem 3: Sumber simplification rule (multi-option games).

Let G = {A, B|C, D}, where A, B, C and D are sumbers. The net weight of C is less than or equal to the net weight of D.

1. If A < | 0, B < | 0 and C | > 0, D | > 0 then G = 0.

2. Otherwise, A ≥ 0, B ≥ 0, C ≤ 0 or D ≤ 0. Without loss of generality, we may assume A ≥ 0 or B ≥ 0 and the net weight of A + C or B + C is greater than or equal to 0 (otherise, apply this rule to −G). By assumption, the net weight of C is greater than or equal to −1.

(a) If C has non-negative net weight then G is a sumber if and only if X(G) is not empty.

(b) If C has net weight −1 and C || 0 then G is a sumber if and only if X(G) is not empty.

(c) Otherwise, C has net weight −1, and C < 0. If A||∗, B||∗, and C||∗, D||∗ then G = ∗.

(d) Otherwise, C has net weight −1, either (A > ∗ or B > ∗) or (C < ∗ or D < ∗). Without loss of generality, we may assume A > ∗ or B > ∗ (otherise, apply this rule to −G). G is a sumber if and only if X(G) is not empty.

In cases 2(a), 2(b), and 2(d), If G is a sumber then G = C_{m} + ∗+ ↑ (p) where m is the minimum cut of C and p is the simplest number in X(G).

The proofs for the above cases are similar to the ones provided in section 3.

5. Examples

Example 4: Consider the game {A|B} = {0| ↑ (1)}

The minimum cut of B occurs at 1. B_{1} =↑ (1) and B^{1} = 0. X(A|B) = {x ≥ 1 : A − B_{1} + ∗ < | ↑ (x) < | B^{1} + ∗} = {x ≥ 1} The simplest number in X(A|B) is 1. By rule 2(a), we have {A|B} = B_{1} + ∗+ ↑ (1) =↑ (1)+ ↑ (1) + ∗. □

Example 4 is the up-star equality first found by Conway[2]. Indeed, the equality still holds if 1 is replaced by any positive integer n.

{0| ↑ (n)} =↑ (n)+ ↑ (n) + ∗.

When n is not an integer, the result is interesting.

Example 5: Consider the game \{A|B\} = \{0| ↑ (\frac{1}{2})\}

The minimum cut of B occurs at \frac{1}{2}. B_{\frac{1}{2}} =↑ (\frac{1}{2}) and B^{\frac{1}{2}} = 0. X(A|B) = \{x ≥ \frac{1}{2} : A − B_{\frac{1}{2}} + ∗ < | ↑ (x) < | B^{\frac{1}{2}} + ∗\} = \{x ≥ \frac{1}{2}\} The simplest number in X(A|B) is 1. By rule 2(a), we have \{A|B\} = B_{\frac{1}{2}} + ∗+ ↑ (1) =↑ (\frac{1}{2})+ ↑ (1) + ∗. □

There are cases where A and B are sumbers with term(s) of integer order(s), but {A|B} is a sumber with term(s ) of non-integer order(s).

Example 6: Consider the game {A|B} = {↑ (1)| ∗ +2. ↑ (2) − 2. ↑ (3)}

The minimum cut of B occurs at 0. B_{0} = ∗ and B^{0} = 2. ↑ (2) − 2. ↑ (3). X(A|B) = {x ≥ 0 : A − B_{0} + ∗ < | ↑ (x) < | B^{0} + ∗} = {1 < x < 2} The simplest number in X(A|B) greater than or equal to 0 is 1\frac{1}{2}. By rule 2(a), we have \{A|B\} = B_{0} + ∗+ ↑ (1\frac{1}{2}) =↑ (1\frac{1}{2}). □

There are cases where A and B are sumbers, but {A|B} is not a sumber. In general, if A − B > ↑ (n) + ∗, for any n, then {A|B} is not a sumber.

Example 7: Consider the game {A|B} = {↑ (1)+ ↑ (1)|∗}

The minimum cut of B occurs at 0. B_{0} = ∗ and B^{0} = 0. X(G) = {x ≥ 0 : A − B_{0} + ∗ < | ↑ (x) < | B^{0} + ∗} = {} X(A|B) is an empty set, so {A|B} is not a sumber. □

There are cases where all the games in G^{L} and G^{R} are sumbers, the difference between G^{L} and G^{R} is less than ↑ (n) + ∗ for some n, but G is not a sumber.

Example 8: Consider the game G = {A|B, C} = {↑ (1)|∗, ↑ (1)}

The minimum cut of B occurs at 0. B_{0} = ∗ and B^{0} = 0. X(G) = {x ≥ 0 : A − B_{0} + ∗ < | ↑ (x) < | B^{0} + ∗ and ↑ (x) < | C − B_{0} + ∗ } = {} X(G) is an empty set, so G is not a sumber. □

6. Conclusion

This article introduces a subgroup of games called sumbers. Similar to numbers and nimbers, sumbers have the following properties:

1. Sumbers are closed under addition.

2. There exists a simple rule to determine the outcome of a sumber.

3. There exists a simplification rule that can, when G is a sumber, determine the exact sumber of G from G^{L} and G^{R}.

(This paper is sponsored by National Science Council, Taiwan. [NSC-93-2213-E-346-006])

References

[1] E. R. Berlekamp, J. H. Conway, R. K. Guy, Winning Ways, Academic Press, New York, 1982.

[2] J. H. Conway, On Numbers and Games, Academic Press, New York, 1976.

[3] C. L Bouton , Nim : A Game With a Complete Mathematical Theory, Annals of Math, Vol. 3, pp. 35-39.

[4] P. M. Grundy, Mathematics and Games, Eureka, 1939.

[5] K. Y. Kao, On Hot and Tepid Combinatorial Games, Ph.D Dissertation, Univ. of North Carolina, Charlotte, 1997.