Table Of ContentTransfinite Version of Welter’s Game
Tomoaki Abuku∗
7
1
0 January 31, 2017
2
n
a
J
1
Abstract
3
Westudythetransfiniteversion ofWelter’s Game, acombinatorial
]
O game, which is played on the belt divided into squares with general
C ordinal numbers extended from natural numbers.
. In particular, we obtain a straight-forward solution for the trans-
h
t finite version based on those of the transfinite version of Nim and the
a
m original version of Welter’s Game.
[
1 1 Introduction
v
8
2
1.1 Impartial game
9
8
0 This paper discusses only “impartial” games, by which we refer to games
.
1 with the following characters:
0
7 • Two players alternately make a move.
1
:
v • No chance elements (the effect of each move can be completely recog-
i
X nized before the move is made).
r
a
• Both players have complete knowledge of the game states.
• The game terminates in finitely many moves.
• Both players have the same set of options of moves in any position.
∗Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba city,
Japan.
1
In general, a game with satisfies the first three characters is called a
combinatorial game.
Combinatorialgamesarecalledinnormal(resp. mis`ere)formiftheplayer
who makes the last move will win (resp. lose).
For the moment, we assume that there are only a finite number of posi-
tions that can be reached from the initial position, and a position may never
be repeated (such a game is called short).
Let G be an impartial game position, we will express the symbol G → G′
means that G′ can be reached by a single move from G.
The original version of Nim, Welter’s GameandGreen Hackenbush which
will be discussed in chapter 1 and 2 are short games. In chapter 3, this
assumption will be removed.
Definition 1.1 (outcome classes). A game position is called an N-position
(resp. a P-position) if the first player (resp. the second player) has a winning
strategy.
Clearly, all impartial game positions are classified into N-positions or
P-positions.
Theorem 1.2 (Bouton[1]). If G is an N-position, there exists a move from
G to a P-position. If G is a P-position, there exists no move from G to a
P-position.
1.2 Nim
Nim is a well-known impartial game with the following rules:
• It is played with several heaps of tokens.
• The legal move is to remove any number of tokens (but necessarily at
least one) from any heap.
• The end position is the state of no heaps of tokens.
Let us denote by N the set of all nonnegative integers.
0
Definition 1.3 (nim-sum). The valueobtained by adding numbers in binary
form without carry is called nim-sum. The nim-sum of nonnegative integers
m ,...,m is written as
1 n
m ⊕···⊕m .
1 n
2
The set N is isomorphic to the direct sum of countably many Z/2Z’s.
0
Also, the nim-sum operation can be extended naturally on Z by using the
2’s complement.
Lemmma 1.4 (Conway[4]). For integer n,
n⊕(−1) = −1−n.
Theorem 1.5 (Bouton[1]). We denote the Nim position with heaps of size
m ,...,m by (m ,...,m ). Then,
1 n 1 n
m ⊕···⊕m 6= 0 ⇐⇒ N-position
1 n
m ⊕···⊕m = 0 ⇐⇒ P-position.
1 n
1.3 Grundy number
Grundy number was introduced in attempt to develop the theory about gen-
eral impartial games, which classifies positions of impartial games such as
Nim.
Definition 1.6 (minimum excluded number mex). Let T be a proper sub-
set of N . Then mex T is defined to be the least nonnegative integer not
0
contained in T, namely
mex T = min(N \T).
0
Definition 1.7 (Grundy number). We denote the end positions by E. Let
G be a game position, and {G′,...,G′ } the set of all positions reached by a
1 n
single move from G. The value G(G) is defined as follows:
0 (G = E)
G(G) =
(cid:26) mex{G(G′),...,G(G′ )} (G 6= E).
1 n
Moreover, G(G) is called the Grundy number of G.
Theorem 1.8 (Sprague[2], Grundy[3]). We have the following for general
short impartial games.
g(G) 6= 0⇐⇒ N-position
g(G) = 0⇐⇒ P-position.
Thanks to this theorem, we only have to decide the Grundy number of
a position for winning strategy in impartial games, and we can classify all
game positions into N-positions or P-positions.
Theorem 1.9 (Grundy[3]). Let (m ,...,m ) be a Nim position, we have
1 n
G(G +···+G ) = m ⊕···⊕m ,
1 n 1 n
where G = G + ··· + G . Refer to [4] for the disjunctive sum + of game
1 n
positions.
3
2 Welter’s Game
2.1 Welter’s Game
Welter’s Game is an impartial game investigated by Welter in 1954. Since it
was also investigated by Mikio Sato, it is often called Sato’s game in Japan.
The rules of Welter’s Games are as follows:
0 1 • • 4 • 6 • 8 9 ···
• It is played with several coins placed on a belt divided into several
squares numbered with the nonnegative integers 0,1,2,... from the
left as shown in the figure above.
• The legal move is to move any one coin from its present square to any
unoccupied square with a smaller number.
• The game terminates when a player is unable to move a coin, namely,
the coins are jammed in squares with the smallest possible numbers as
shown in the figure below.
• • • • 4 5 6 7 8 9 ···
This game is equivalent to Nim with an additional rule that you cannot
make two heaps with the same number of tokens.
Sincethisgameisashort impartialgame, ouraimistodecidetheGrundy
number of a position.
2.2 Welter function
When an expression in what follows includes both nim-sum and the four
basic operations of arithmetic without parentheses, we will make it a rule to
calculate nim-sums prior to the others, and we express the nim-summation
⊕
by the symbol .
X
Definition 2.1 (mating function). Mating function (x | y) is defined by
2n+1 −1 (x ≡ y (mod 2n), x 6≡ y (mod 2n+1))
(x | y) =
(cid:26) −1 (x = y).
Particularly, if x and y have different parities, then (x | y) = 1.
Then we have the following:
4
(x | y) = (x−y)⊕(x−y−1), and (x | y) = (x+a | y+a) = (x⊕a | y⊕a).
Lemmma 2.2 (Conway[4]). For x,y,z ∈ Z, we have the following equalities:
x⊕(x | 0) = x−1y ⊕(y | −1) = y +1x⊕y ⊕(x | y) = x⊕y −1.
Lemmma 2.3 (Conway[4]). For distinct integers x,y,z ∈ Z, two of (x |
y),(x | z),(y | z) are equal and the rest is greater than them
Definition 2.4 (animatingfunction). Foranynonnegativeintegersa,b,c,d,...,
a function f(x) of form
f(x) = (((x⊕a)+b)⊕c)+d⊕···
is called an animating function.
If f and g are animating functions, f(g(x)) and f−1(x) are clearly ani-
mating functions. Also, we have f−1(x) = ((((···x···)−d)⊕)−b)⊕a. Thus,
the set of all animating functions forms a group with respect to composition.
Theorem 2.5 (Conway[4]). Any animating function can be written as
f(x) = x⊕n⊕(x | p )⊕···⊕(x | p )
1 k
for some n,p ,...,p ∈ Z. Conversely, function the form of above is an
1 k
animating function.
Definition 2.6 (Welter function). Let (a ,...,a ) be a Welter’s Game posi-
1 n
tion. Then we define the value [a |···|a ] of Welter function at (a ,...,a )
1 n 1 n
as follows:
⊕
[a |···|a ] = a ⊕···⊕a ⊕ (a | a ).
1 n 1 n i j
1≤Xi<j≤n
In the case of one coin, clearly [a ] = a . In the case of two coins
1 1
[a |a ] = a ⊕a ⊕(a | a ) = a ⊕a −1.
1 2 1 2 1 2 1 2
Let (a ,...,a ) be a position in Welter’s Game and a , a the pair with
1 n i j
the largest mating function value (a | a ) (that is, a and a are congruent
i j i j
to each other modulo the highest possible power of 2 among all pairs).
5
By Lemma 2.3 mating function values (a | a ) and (a | a ) cancel each
i k j k
other for all other a ’s. Thus, for example, if i = 1, j = 2, we can calculate
k
the Welter function as follows:
⊕
[a |···|a ] = a ⊕···⊕a ⊕ (a | a )
1 n 1 n i j
1≤Xi<j≤n
⊕
= a ⊕a ⊕(a | a )⊕ (a | a )
1 2 1 2 1 k
3≤Xk≤n
⊕
⊕ (a | a )⊕a ⊕···⊕a
2 k 3 n
3≤Xk≤n
⊕
⊕ (a | a )
i j
3≤Xi<j≤n
= a ⊕a ⊕(a | a )⊕a ⊕···⊕a
1 2 1 2 3 n
⊕
⊕ (a | a )
i j
3≤Xi<j≤n
= [a |a ]⊕[a |a |···].
1 2 3 4
This method is called splitting.
Lemmma 2.7 (Conway[4]). a > a′,a > a′,a > a′,... are legal moves in
1 1 2 2 3 3
Welter’s Game, we have the following:
[a′|a |a |···] = [a |a′|a |···] ⇐⇒ [a′|a′|a |···] = [a |a |a |···].
1 2 3 1 2 3 1 2 3 1 2 3
Theorem 2.8 (Welter’s Theorem[5]). The value of Welter function at each
position in Welter’s Game is equal to its Grundy number in Welter’s Game.
Namely, we have the following:
G(a ,...,a ) = [a |...|a ].
1 n 1 n
2.2.1 Mating Method
Splitting reduces calculation of Welter function to that of the two argument
Welter function and nim-sum.
When we mate pairs with the largest mating function value in orderwe
have the following equality using splittingFor Welter function of n arguments
[a |a ]⊕[a |a ]⊕··· (n : even)
1 2 3 4
[a |···|a ] =
1 n (cid:26) [a1|a2]⊕[a3|a4]⊕···⊕[an] (n : odd),
6
where (a | a ),(a | a ),... is arranged in order of the values of mating
1 2 3 4
function.
By using this equality and formulas [a ] = a and [a |a ] = a ⊕a −1,
1 1 1 2 1 2
we can easily compute the value of Welter function This method is called
Mating Method.
Welter function is an animating function with respect to each of its ar-
guments. Since an animating function is a bijection on Z, for any integers
a ,...,a and s, there exists a unique integer solution x for equation
1 n
[x | a |···|a ] = s.
1 n
Moreover if a ,...,a are distinct nonnegative integers, and s is nonneg-
1 n
ative, x is a nonnegative integer distinct from a ,...,a .
1 n
3 Transfinite Game
3.1 Transfinite Nim
First, weextendNimintoitstransfiniteversion(TransfiniteNim)byallowing
thesizeoftheheaps oftokens tobeageneralordinalnumber. Thelegalmove
istoreplaceanarbitraryordinalnumber αbyasmaller numberβ. Therefore,
Transfinite Nim may not necessarily be short.
Let us denote by ON the class of all ordinal numbers. Later we see that
the nim-sum operation can be extended naturally on ON.
The following is known about general ordinal numbers.
Theorem 3.1 (Cantor Normal Form theorem[6]). Every α ∈ ON(α > 0)
can be expressed as
α = ωγk ·m +···+ωγ1 ·m +ωγ0 ·m ,
k 1 0
where k is a nonnegative integer, m ,...,m ∈ N \{0}, and α ≥ γ > ··· >
0 k 0 k
γ > γ ≥ 0.
1 0
Let α ,...,α be ordinal numbers. Then, each α , i = 1,...,n is ex-
1 n i
pressed by using finite by many common powers γ ,...,γ as:
0 k
α = ωγk ·m +···+ωγ1 ·m +ωγ0 ·m ,
i ik i1 i0
where m ∈ N .
ik 0
Next, we will define the minimal excluded number of a set of ordinals and
the Grundy number of a position in general Transfinite Game.
7
Definition 3.2 (minimalexcludednumbermex). LetT beapropersubclass
of ON. Then mex T is defined to be the least ordinal number not contained
in T, namely
mex T = min(ON \T).
Definition 3.3 (Grundy number). Let G be an impartial game (it may not
necessarily be short). The value G(G) is defined as
G(G) = mex{G(G′) | G → G′}.
Theorem 3.4. We have the following for Transfinite impartial games:
G(G) 6= 0⇐⇒ N-position
G(G) = 0⇐⇒ P-position.
Definition 3.5. For ordinal numbers α ,...,α ∈ ON, we define their nim-
1 n
sum as follows:
α ⊕···⊕α = ωγk(m ⊕···⊕m ).
1 n 1k nk
X
k
Theorem 3.6. For Transfinite Nim position (α ,...,α ) ⊆ ONnwe have
1 n
the following:
G(α ,...,α ) = α ⊕···⊕α .
1 n 1 n
Proof. The proof is by induction. Let α ⊕···⊕α = α (α ∈ ON). We have
1 n
to show that, for each β (< α), there exists a position reached by a single
move from (α ,...,α ) and that its Grundy number is β.
1 n
Let (α ,...,α ) → (β ,...,β ), by induction assumption we have
1 n 1 n
G(β ,...,β ) = β ⊕···⊕β .
1 n 1 n
If α = 0, no ordinal β (β < α) exists. We can assume α > 0.
We can write α and β as
α = ωγk ·a +···+ωγk ·a +a
k 1 0
β = ωγk ·b +···+ωγk ·b +b ,
k 1 0
where a ,...,a ,b ,...,b ∈ N . By definition,
0 k 0 k 0
a = m ⊕···⊕m , for s = 1,...,k.
s 1s ns
Since α > β, there exsists s such that
a > b , a = b for all t (< s).
s s t t
8
As in the strategy of original Nim, since a > b , there is an index i such
s s
that
m > m ⊕a ⊕b .
is is s s
We define
m′ = m ⊕a ⊕b for all t (≤ s)
it it s s
and
α′ = ωγk ·m +···ωγs+1 ·m +ωγs ·m′
i ik is+1 is
+ωγs−1 ·m′ +···+ωγ0 ·m′ ,
is−1 i0
where m ⊕a ⊕b = m′ .
is s s is
If we put α′ = β , α = β (j 6= i). Then, α > β and we have
i i j j i i
β ⊕···β ⊕β ⊕β ⊕···β = β
1 i−1 i i+1 n
Therefore, for each β (< α), there is a position (β ,...,β ) reached by a
1 n
single move from (α ,...,α ).
1 n
Example 3.7. In the case of position (1,ω·2+4,ω2·3+9,ω2·2+ω ·4+
16,ω2 +ω ·5+25):
Let us calculate the value of α ⊕α ⊕α ⊕α ⊕α .
1 2 3 4 5
We get
α = ωβ2 ·m +ωβ1 ·m +m = ω2 ·0+ω ·0+1
1 12 11 10
α = ωβ2 ·m +ωβ1 ·m +m = ω2 ·0+ω ·2+4
2 22 21 20
α = ωβ2 ·m +ωβ1 ·m +m = ω2 ·3+ω ·0+9
3 32 31 30
α = ωβ2 ·m +ωβ1 ·m +m = ω2 ·2+ω ·4+16
4 42 41 40
α = ωβ2 ·m +ωβ1 ·m +m = ω2 ·1+ω ·5+25.
5 52 51 50
So, we have
m ⊕m ⊕m ⊕m ⊕m = 0⊕0⊕3⊕2⊕1
12 22 32 42 52
= 0
m ⊕m ⊕m ⊕m ⊕m = 0⊕2⊕0⊕4⊕5
11 21 31 41 51
= 3
m ⊕m ⊕m ⊕m ⊕m = 1⊕4⊕9⊕16⊕25
10 20 30 40 50
= 5.
Thus, by the definition of nim-sum in general ordinal number
α ⊕α ⊕α ⊕α ⊕α = ω ·3+5.
1 2 3 4 5
Therefore, this position is an N-position, and the legal good move is
ω ·2+4 → ω +1.
9
3.2 Transfinite Welter’s Game
In Transfinite versionthe size of the belt of Welter’s Game is extended into
general ordinal numbers, but played with finite number of coins. The legal
moveistomoveonecointowardtheleft(jumpingisallowed), andyoucannot
place two or more coins on the same square as in the original Welter’s Game.
0 1 • 3 ··· ω • ω +2 ··· ω2 ···
Definition 3.8. Let α ,...,α ∈ ON. Each α can be expressed as α =
1 n i i
ω·λ +m , where λ ∈ ON and m ∈ N . Welter function in general ordinal
i i i i 0
numbers is defined as follows:
⊕
[α |···|α ] = ω ·(λ ⊕···⊕λ )+ [S ],
1 n 1 n λ
λX∈ON
where [S ] is Welter function, and S = {m | λ = λ}.
λ λ n n
We obtain the following main theorem.
Theorem 3.9. Let α ,...,α ∈ ON. Grundy number of general position
1 n
(α ,...,α ) in Transfinite Welter’s Game is equal to its Welter function.
1 n
Namely, we have the following:
G(α ,...,α ) = [α |···|α ].
1 n 1 n
Proof. Let [α |···|α ] = α. We have to show that, for each β (< α), there
1 n
exists a position with Grundy number β which is reached by a single move
from (α ,...,α ).
1 n
Let (α ,...,α ) → (β ,...,β ). Then by the assumption of induction we
1 n 1 n
have
G(β ,...,β ) = [β |···|β ].
1 n 1 n
If α = 0, there exist no β (< α). We can assume α > 0 and
α = ω ·λ+a and β = ω ·λ′ +b ,
0 0
where λ, λ′ ∈ ON, a , b ∈ N . Since α > βwe have
0 0 0
(λ > λ′) or (λ = λ′ and a > b ).
0 0
⊕
In the latter case, since a = [S ] > b , from theory of Nim[1][2][3],
0 λ 0
λX∈ON
there exists some λ and nonnegative integer c (< [S ]) such that
0 0 λ0
a ⊕[S ]⊕c = b .
0 λ0 0 0
10
