We shall study the following restricted Wythoff’s Nim. Let Si (1 ≤ i ≤ 3) be the set of positive integers. Each player can remove the number of tokens s1 ∈ S1 from the first pile and s2 ∈ S2 from the second pile and remove the same number of tokens s3 ∈ S3 from both piles. We shall identify (m, n) to a position of this nim, where m is the number of tokens in the first pile and n is the number of tokens in the second pile. In the case |S2| < ∞, we will show the Sprague-Grundy sequence(or simply G-sequences) gS(m, n) is periodic for fixed m

Katayama, Shin-ichi

Kubo, Tomoya

Tokushima University Institutional Repository

53J. Math. Tokushima Univ. Vol. 52 (2018), 53−57[2] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoﬀ string,Trans. Amer. Math. Soc. 348 (1996) 305–330.[3] G.F. Carrier, On the non-linear vibration problem of the elastic string,Quart. Appl. Math. 3 (1945) 157–165.[4] R.W. Dickey, Inﬁnite systems of nonlinear oscillation equations with lineardamping, SIAM J. Appl. Math. 19 (1970) 208–214.[5] M. Ghisi and M. Gobbino, Hyperbolic-parabolic singular perturbation formildly degenerate Kirchhoﬀ equations: time-decay estimates, J. Diﬀeren-tial Equations 245 (2008) 2979–3007.[6] G. Kirchhoﬀ, Vorlesungen u¨ber Mechanik, Teubner, Leipzig, 1883.[7] K. Ono, Global existence and decay properties of solutions for some mildlydegenerate nonlinear dissipative Kirchhoﬀ strings, Funkcial. Ekvac. 40(1997) 255–270.[8] K. Ono, Decay estimates of solutions for mildly degenerate Kirchhoﬀ typedissipative wave equations in unbounded domains Asymptot. Anal. 88(2014) 75–92.[9] K. Ono, Lower decay estimates for non-degenerate dissipative wave equa-tions of Kirchhoﬀ type Sci. Math. Japonicae 77 (2014) 415–425.[10] K. Ono, Asymptotic behavior of solutions for Kirchhoﬀ type dissipativewave equations in unbounded domains J. Math. Tokushima Univ., 61(2017) 37–54.[11] W.A. Strauss, On continuity of functions with values in various Banachspaces, Paciﬁc J. Math. 19 (1966) 543–551.14On Restricted Wythoﬀ’s NimByShin-ichi Katayama and Tomoya KuboShin-ichi KatayamaDepartment of Mathematical Sciences, Graduate School of Science and Technology,Tokushima University, Minamijosanjima-cho 2-1, Tokushima 770-8506, JAPANe-mail address : shinkatayama@tokushima-u.ac.jpandTomoya KuboGraduate School of Integrated Arts and Sciences, Tokushima University,Minamijosanjima-cho 1-1, Tokushima 770-8502, JAPANe-mail address : itiji banji@me.comReceived October 12 2018AbstractWe shall study the following restricted Wythoﬀ’s Nim. LetSi (1 ≤ i ≤ 3) be the set of positive integers. Each player canremove the number of tokens s1 ∈ S1 from the ﬁrst pile and s2 ∈ S2from the second pile and remove the same number of tokens s3 ∈ S3from both piles. We shall identify (m,n) to a position of this nim,where m is the number of tokens in the ﬁrst pile and n is the num-ber of tokens in the second pile. In the case |S2| <∞, we will showthe Sprague-Grundy sequence(or simply G-sequences) gS(m,n) isperiodic for ﬁxed m.2010 Mathematics Subject Classiﬁcation. Primary 91A46; Sec-ondary 91A051 IntroductionIn his paper [1], C. L. Bouton introduced the 2-player impartial combinatorialgame, which is now called nim game. In [6], W. A. Wythoﬀ modiﬁed the rule1On Restricted Wythoﬀ’s NimByShin-ichi Katayama and Tomoya KuboShin-ichi KatayamDepartment of Mathematic l Scien es, Gradu te School f Scien e and Technology,Tokushima University, Minamijosanjima-cho 2-1, Tokushima 770-8506, JAPANe-mail address : shinkatayama@tokushima-u. c.jpandTom ya KuboGradu te School f Integrated Arts and Scien es, Tokushima University,Minamijosanjima-cho 1-1, Tokushima 770-8502, JAPANe-mail address : itiji banji@me.comReceived October 12 2018AbstractWe shall study the following rest icted Wythoﬀ’s Nim. LetSi (1 ≤ i ≤ 3) be the set of positive ntegers. Each player canremove th number of tokens s1 ∈ S1 from the ﬁrst pile and s2 ∈ S2from the second pile and remove th same number of tokens s3 ∈ S3from both piles. We shall identify (m,n) to a position of this nim,where m is the number of t kens i the ﬁrst pile and n is the num-ber of tokens i the second pile. In the case |S2| <∞, we ill showthe Sprague-Grundy sequ nce(or simply G-sequ nces) gS(m,n) isperiodic for ﬁxed m.2010 Mathematics Subject Classiﬁcation. Primary 91A46; Sec-ondary 91A051 IntroductionIn his paper [1], C. L. Bouton introduced the 2-player imparti l combinatorialgame, which s now called nim game. In [6], W. A. Wythoﬀ modiﬁe the rule1Shin-ichi Katayama and Tomoya Kubo54of this game as follows. The game is played by two players. There are two pilesof tokens(or stones). Two players play alternately and either take from one ofthe piles an arbitrary number of tokens or from both piles of the same numberof tokens. The player who takes up the last token is the winner. A positionfrom which the player who made the last move, the previous player, can alwayswin is called a P-position. The P-positions of original nim of Bouton are (k, k)with arbitrary k ≥ 0 and the following P-positions of Wythoﬀ’s nim are relatedto the golden ratio.Proposition 1.1 Wythoﬀ (1905)(m,n) is a P-position ⇐⇒ (m,n) = (ms,ms + s) or (ms + s,ms), where msis determined by ms = [sϕ]. Here ϕ =1 +√52.In this paper, we shall consider some restricted Wythoﬀ’s nim as follows. LetSi (1 ≤ i ≤ 3) be the set of positive integers. Each player can remove thenumber of tokens s1 ∈ S1 from the ﬁrst pile and s2 ∈ S2 from the second pileand remove the same number of tokens s3 ∈ S3 from both piles. We shallidentify (m,n) to a position of this Nim, where m is the number of tokens inthe ﬁrst pile and n is the number of tokens in the second pile. Assume the setof positive integers S2 is ﬁnite. In the next section, we shall show the Sprague-Grundy sequence gS(m,n) is periodic for ﬁxed m, i.e., there exist am ≥ 0 andpm > 0 such that gs(m,n+ pm) = gs(m,n) for any n ≥ am. Here pm is calledthe period of this Sprague-Grundy sequence gs(m,n).2 Proof of Main TheoremLet S2 = {s1, s2, . . . , ss(2) | 0 < s1 < s2 < · · · < ss(2)} be the set of positiveintegers. The player is restricted to remove the number of tokens s ∈ S2 fromthe second pile.Theorem 2.1 Under the above notations, gS(m,n) has a period pm for anyfixed m, that is,n ≥ am =⇒ gS(m,n+ pm) = gS(m,n), for any n ≥ am.Proof. From the assumption on S2, G-sequence satisﬁes 0 ≤ gS(m,n) ≤ 2m+s(2). The case m = 0 is nothing but the case of restricted one pile nim andit is known that gS(0, n) is periodic. Thus, assume gS(m′, n+ pm′) = g(m′, n)for any n ≥ am′ for the cases m′(0 ≤ m′ ≤ m − 1). p0, p1, . . . , pm−1 denotethe periods for the cases 0 ≤ m′ ≤ m − 1. Put a∗ = max{a0, a1, . . . , am−1},p∗ = LCM(p0, p1, . . . , pm−1). Then the pigeonhole principle asserts that thereexists a period p = pm (p∗|pm) as follows.2The number of patterns of consecutive s(2) Grundy numbers gS(m,n) are atmost ℓ∗ = (2m+ s(2)+ 1)s(2). Consider ℓ∗ +1 = (2m+ s(2)+ 1)s(2)+1 pairs;(gS(m, a∗), gS(m, a∗ + 1), . . . , gS(m, a∗ + s(2)− 1)),(gS(m, a∗ + p∗), gS(m, a∗ + p∗ + 1), . . . , gS(m, a∗ + p∗ + s(2)− 1)),...(gS(m, a∗ + ℓ∗p∗), gS(m, a∗ + ℓ∗p∗ + 1), . . . , gS(m, a∗ + ℓ∗p∗ + s(2)− 1)).Hence the pigeonhole principle asserts that there exists a pair ℓi, ℓj (0 ≤ ℓi <ℓj ≤ ℓ∗) which satisﬁes(gS(m, a∗ + ℓip∗), gS(m, a∗ + ℓip∗ + 1), . . . , gS(m, a∗ + ℓip∗ + s(2)− 1))= (gS(m, a∗ + ℓjp∗), gS(m, a∗ + ℓjp∗ + 1), . . . , gS(m, a∗ + ℓjp∗ + s(2)− 1))Put a = a∗ + p∗ℓi and p = p∗(ℓj − ℓi). From the above condition,gS(m, a) = gS(m, a+ p).........gS(m, a+ s(2)− 1) = gS(m, a+ s(2)− 1 + p)Thus gS(m,n) has period p for a ≤ n ≤ a + s(2) − 1. Assume gS(m,n′) =gS(m,n′ + p) for any n′ (a ≤ n′ < n). Since n − sj ≥ a, n − sk ≥ a,gS(m,n+ p) = mex{gS(m− si, n+ p), gS(m,n+ p− sj), gS(m− sk, n+ p− sk)| si, sj , sk ∈ S with 0 ≤ m− si and 0 ≤ m− sk}= mex{gS(m−si, n), gS(m,n−sj), gS(m − sk, n − sk) | si, sj , sk ∈ S with 0 ≤ m− si and 0 ≤ m− sk} =gS(m,n)．Hence, by induction, we have n ≥ a =⇒ gS(m,n+ p) = gS(m,n).Now we shall consider the special case when |S1|, |S2| and |S3| are ﬁnite. PutS1 = {s1,1, s1,2, . . . , s1,r(1) | 0 < s1,1 < s1,2 < · · · < s1,r(1)},S2 = {s2,1, s2,2, . . . , s2,r(2) | 0 < s2,1 < s2,2 < · · · < s2,r(2)}，and S3 = {s3,1, s3,2, . . . , s3,r(3) | 0 < s3,1 < s3,2 < · · · < s3,r(3)}. s(0) denotesmax(s1,r(1), s2,r(2), s3,r(3)). Assume that there exist a positive integer p whichsatisﬁesgS(m,n+ p) = gS(m,n) for any 0 ≤ m,n ≤ s(0) + p.Then we have the following special case of the above theorem.Corollary 2.2 Under the above notation, gS(m,n) is purely periodic and sat-isfies gS(m,n+ p) = gS(m,n) for any m,n ≥ 0.Using this corollary, we have the following example.3On Restricted Wythoff’s Nim 55of this game as follows. The game is played by two players. There are two pilesof tokens(or stones). Two players play alternately and either take from one ofthe piles an arbitrary number of tokens or from both piles of the same numberof tokens. The player who takes up the last token is the winner. A positionfrom which the player who made the last move, the previous player, can alwayswin is called a P-position. The P-positions of original nim of Bouton are (k, k)with arbitrary k ≥ 0 and the following P-positions of Wythoﬀ’s nim are relatedto the golden ratio.Proposition 1.1 Wythoﬀ (1905)(m,n) is a P-position ⇐⇒ (m,n) = (ms,ms + s) or (ms + s,ms), where msis determined by ms = [sϕ]. Here ϕ =1 +√52.In this paper, we shall consider some restricted Wythoﬀ’s nim as follows. LetSi (1 ≤ i ≤ 3) be the set of positive integers. Each player can remove thenumber of tokens s1 ∈ S1 from the ﬁrst pile and s2 ∈ S2 from the second pileand remove the same number of tokens s3 ∈ S3 from both piles. We shallidentify (m,n) to a position of this Nim, where m is the number of tokens inthe ﬁrst pile and n is the number of tokens in the second pile. Assume the setof positive integers S2 is ﬁnite. In the next section, we shall show the Sprague-Grundy sequence gS(m,n) is periodic for ﬁxed m, i.e., there exist am ≥ 0 andpm > 0 such that gs(m,n+ pm) = gs(m,n) for any n ≥ am. Here pm is calledthe period of this Sprague-Grundy sequence gs(m,n).2 Proof of Main TheoremLet S2 = {s1, s2, . . . , ss(2) | 0 < s1 < s2 < · · · < ss(2)} be the set of positiveintegers. The player is restricted to remove the number of tokens s ∈ S2 fromthe second pile.Theorem 2.1 Under the above notations, gS(m,n) has a period pm for anyfixed m, that is,n ≥ am =⇒ gS(m,n+ pm) = gS(m,n), for any n ≥ am.Proof. From the assumption on S2, G-sequence satisﬁes 0 ≤ gS(m,n) ≤ 2m+s(2). The case m = 0 is nothing but the case of restricted one pile nim andit is known that gS(0, n) is periodic. Thus, assume gS(m′, n+ pm′) = g(m′, n)for any n ≥ am′ for the cases m′(0 ≤ m′ ≤ m − 1). p0, p1, . . . , pm−1 denotethe periods for the cases 0 ≤ m′ ≤ m − 1. Put a∗ = max{a0, a1, . . . , am−1},p∗ = LCM(p0, p1, . . . , pm−1). Then the pigeonhole principle asserts that thereexists a period p = pm (p∗|pm) as follows.2The number of patterns of consecutive s(2) Grundy numbers gS(m,n) are atmost ℓ∗ = (2m+ s(2)+ 1)s(2). Consider ℓ∗ +1 = (2m+ s(2)+ 1)s(2)+1 pairs;(gS(m, a∗), gS(m, a∗ + 1), . . . , gS(m, a∗ + s(2)− 1)),(gS(m, a∗ + p∗), gS(m, a∗ + p∗ + 1), . . . , gS(m, a∗ + p∗ + s(2)− 1)),...(gS(m, a∗ + ℓ∗p∗), gS(m, a∗ + ℓ∗p∗ + 1), . . . , gS(m, a∗ + ℓ∗p∗ + s(2)− 1)).Hence the pigeonhole principle asserts that there exists a pair ℓi, ℓj (0 ≤ ℓi <ℓj ≤ ℓ∗) which satisﬁes(gS(m, a∗ + ℓip∗), gS(m, a∗ + ℓip∗ + 1), . . . , gS(m, a∗ + ℓip∗ + s(2)− 1))= (gS(m, a∗ + ℓjp∗), gS(m, a∗ + ℓjp∗ + 1), . . . , gS(m, a∗ + ℓjp∗ + s(2)− 1))Put a = a∗ + p∗ℓi and p = p∗(ℓj − ℓi). From the above condition,gS(m, a) = gS(m, a+ p).........gS(m, a+ s(2)− 1) = gS(m, a+ s(2)− 1 + p)Thus gS(m,n) has period p for a ≤ n ≤ a + s(2) − 1. Assume gS(m,n′) =gS(m,n′ + p) for any n′ (a ≤ n′ < n). Since n − sj ≥ a, n − sk ≥ a,gS(m,n+ p) = mex{gS(m− si, n+ p), gS(m,n+ p− sj), gS(m− sk, n+ p− sk)| si, sj , sk ∈ S with 0 ≤ m− si and 0 ≤ m− sk}= mex{gS(m−si, n), gS(m,n−sj), gS(m − sk, n − sk) | si, sj , sk ∈ S with 0 ≤ m− si and 0 ≤ m− sk} =gS(m,n)．Hence, by induction, we have n ≥ a =⇒ gS(m,n+ p) = gS(m,n).Now we shall consider the special case when |S1|, |S2| and |S3| are ﬁnite. PutS1 = {s1,1, s1,2, . . . , s1,r(1) | 0 < s1,1 < s1,2 < · · · < s1,r(1)},S2 = {s2,1, s2,2, . . . , s2,r(2) | 0 < s2,1 < s2,2 < · · · < s2,r(2)}，and S3 = {s3,1, s3,2, . . . , s3,r(3) | 0 < s3,1 < s3,2 < · · · < s3,r(3)}. s(0) denotesmax(s1,r(1), s2,r(2), s3,r(3)). Assume that there exist a positive integer p whichsatisﬁesgS(m,n+ p) = gS(m,n) for any 0 ≤ m,n ≤ s(0) + p.Then we have the following special case of the above theorem.Corollary 2.2 Under the above notation, gS(m,n) is purely periodic and sat-isfies gS(m,n+ p) = gS(m,n) for any m,n ≥ 0.Using this corollary, we have the following example.3Shin-ichi Katayama and Tomoya Kubo56Table 1 (The case S = ({1, 2}, {1, 2}, {1, 2}))m\n 0 1 2 3 4 5 6 7 80 0 1 2 0 1 2 0 1 21 1 2 0 1 2 0 1 2 02 2 0 1 2 0 1 2 0 13 0 1 2 0 1 2 0 1 24 1 2 0 1 2 0 1 2 05 2 0 1 2 0 1 2 0 1Thus we have gS(m,n) ≡ g(m,n) (mod 3) and gS(m,n + 3) = gS(m,n) forany m and n.3 Equivalent classes of S ⊂ NWe call S ⊂ N and S′ ⊂ N is equivanent if and only ifS ∼ S′ ⇐⇒ gS(k) = gS′(k) (for any k ∈ N0 = {0} ∪ N).For n piles (m1,m2, . . . ,mn), the number of removable tokens from the pilemk is restricted s ∈ Sk for each Sk of S = (S1, S2, . . . , Sk, . . . , Sn). Then weshall slightly generalizes the equivalent classes for S = (S1, S2, . . . , Sn), andS′ = (S′1, S′2, . . . , S′n),S ∼ S′ ⇐⇒ gS(k1, k2, . . . , kn) = gS′(k1, k2, . . . , kn) (for any ki ∈ N0 (1 ≤ i ≤ n)).Then nim-sum impliesS ∼ S′ ⇐⇒ Si ∼ S′i (1 ≤ i ≤ n).For any Si, S′i ⊂ N (1 ≤ i ≤ 3), we shall write S = (S1, S2, S3), S′ = (S′1, S′2, S′3)and consider the restricted Wythoﬀ’s nim. In the case S, the number of remov-able tokens from the ﬁrst pile is restricted to s ∈ S1, the number of removabletokens from the second pile is restricted to s ∈ S2, and the number of removabletokens from both piles at the same time is restricted to s ∈ S3. The case S′is same as the case S. Now we will consider two restricted Wythof’s nim suchas the number of tokens each player can remove from the piles are restricteds ∈ S and s ∈ S′. Then for these restricted Wythoﬀ’s nim, there exist severalS and S′ with S ∼ S′ but gS(m,n) ̸= gS′(m,n) for some (m,n).Put S = 2N and S′ = N − {1}. Then it is known that S ∼ S′. We havecalculated gS(m,n) gS′(m,n) for small (m,n) as follows.4Table 2-1 (The case S = 2N) Table 2-2 (The case S′ = N− {1})m\n 0 1 2 3 4 5 6 70 0 0 1 1 2 2 3 31 0 0 1 1 2 2 3 32 1 1 2 2 0 0 4 43 1 1 2 2 0 0 4 44 2 2 0 0 1 1 5 55 2 2 0 0 1 1 5 5m\n 0 1 2 3 4 5 6 70 0 0 1 1 2 2 3 31 0 0 1 1 2 2 3 32 1 1 2 2 0 0 4 43 1 1 2 2 3 0 0 44 2 2 0 3 1 4 5 55 2 2 0 0 4 1 5 5Then one can verify that S ∼ S ′, but gS(3, 4) = 0, gS′(3, 4) = 3.Put (S = ({1, 2, 3}, {1, 2, 3}, {1, 2, 3}) and S′ = ({1, 2, 3}, {1, 2, 3, 5}, {1, 2, 3})).There are some examples gS(m,n) gS′(m,n) for small (m,n) as follows.Table 3-1 Table 3-2(S = ({1, 2, 3}, {1, 2, 3}, {1, 2, 3})) (S′ = ({1, 2, 3}, {1, 2, 3, 5}, {1, 2, 3}))m\n 0 1 2 3 4 5 6 70 0 1 2 3 0 1 2 31 1 2 0 4 1 2 0 42 2 0 1 5 3 0 1 23 3 4 5 6 2 7 4 5m\n 0 1 2 3 4 5 6 70 0 1 2 3 0 1 2 31 1 2 0 4 1 2 0 42 2 0 1 5 3 0 1 23 3 4 5 6 2 7 4 6One can also verify that S ∼ S ′, but gS(3, 7) = 5, gS′(3, 7) = 6 for this case.References[ 1 ] C. L. Bouton, Nim, A Game with a Complete Mathematical Theory,Ann. of Math., 3 (1901–1902), 35–39.[ 2 ] A. Dress, A. Flammenkamp and N. Pink, Additive Periodicity of TheSprague-Grundy Function of Certain Nim Games, Adv. Appl. Math. 22(1999), 249–270.[ 3 ] S. Katayama and T. Kubo, Wythoﬀ’s Stone up Game and the Theoremof Rayleigh, Journal of the Tokushima Society for the History of Science37 (2018), to appear.[ 4 ] W. A. Liu, H. Li and B. Li, A Restricted Version of Wythoﬀ’s Nim,The Electric Journal of Combinatrics 18 (2011), # p 207.[ 5 ] F. Sato, On the Mathematics of Extracting Games of Stones-WonderfulRelations between Games and Algebra, Sugaku-Shobou, 2014 (in Japanese)[ 6 ] W. A. Wythoﬀ, A Modiﬁcation of the Game of Nim, Nieuw Arch, Wisk.7 (1907), 199–202.5On Restricted Wythoff’s Nim 57Table 1 (The case S = ({1, 2}, {1, 2}, {1, 2}))m\n 0 1 2 3 4 5 6 7 80 0 1 2 0 1 2 0 1 21 1 2 0 1 2 0 1 2 02 2 0 1 2 0 1 2 0 13 0 1 2 0 1 2 0 1 24 1 2 0 1 2 0 1 2 05 2 0 1 2 0 1 2 0 1Thus we have gS(m,n) ≡ g(m,n) (mod 3) and gS(m,n + 3) = gS(m,n) forany m and n.3 Equivalent classes of S ⊂ NWe call S ⊂ N and S′ ⊂ N is equivanent if and only ifS ∼ S′ ⇐⇒ gS(k) = gS′(k) (for any k ∈ N0 = {0} ∪ N).For n piles (m1,m2, . . . ,mn), the number of removable tokens from the pilemk is restricted s ∈ Sk for each Sk of S = (S1, S2, . . . , Sk, . . . , Sn). Then weshall slightly generalizes the equivalent classes for S = (S1, S2, . . . , Sn), andS′ = (S′1, S′2, . . . , S′n),S ∼ S′ ⇐⇒ gS(k1, k2, . . . , kn) = gS′(k1, k2, . . . , kn) (for any ki ∈ N0 (1 ≤ i ≤ n)).Then nim-sum impliesS ∼ S′ ⇐⇒ Si ∼ S′i (1 ≤ i ≤ n).For any Si, S′i ⊂ N (1 ≤ i ≤ 3), we shall write S = (S1, S2, S3), S′ = (S′1, S′2, S′3)and consider the restricted Wythoﬀ’s nim. In the case S, the number of remov-able tokens from the ﬁrst pile is restricted to s ∈ S1, the number of removabletokens from the second pile is restricted to s ∈ S2, and the number of removabletokens from both piles at the same time is restricted to s ∈ S3. The case S′is same as the case S. Now we will consider two restricted Wythof’s nim suchas the number of tokens each player can remove from the piles are restricteds ∈ S and s ∈ S′. Then for these restricted Wythoﬀ’s nim, there exist severalS and S′ with S ∼ S′ but gS(m,n) ̸= gS′(m,n) for some (m,n).Put S = 2N and S′ = N − {1}. Then it is known that S ∼ S′. We havecalculated gS(m,n) gS′(m,n) for small (m,n) as follows.4Table 2-1 (The case S = 2N) Table 2-2 (The case S′ = N− {1})m\n 0 1 2 3 4 5 6 70 0 0 1 1 2 2 3 31 0 0 1 1 2 2 3 32 1 1 2 2 0 0 4 43 1 1 2 2 0 0 4 44 2 2 0 0 1 1 5 55 2 2 0 0 1 1 5 5m\n 0 1 2 3 4 5 6 70 0 0 1 1 2 2 3 31 0 0 1 1 2 2 3 32 1 1 2 2 0 0 4 43 1 1 2 2 3 0 0 44 2 2 0 3 1 4 5 55 2 2 0 0 4 1 5 5Then one can verify that S ∼ S′, but gS(3, 4) = 0, gS′(3, 4) = 3.Put (S = ({1, 2, 3}, {1, 2, 3}, {1, 2, 3}) and S′ = ({1, 2, 3}, {1, 2, 3, 5}, {1, 2, 3})).There are some examples gS(m,n) gS′(m,n) for small (m,n) as follows.Table 3-1 Table 3-2(S = ({1, 2, 3}, {1, 2, 3}, {1, 2, 3})) (S′ = ({1, 2, 3}, {1, 2, 3, 5}, {1, 2, 3}))m\n 0 1 2 3 4 5 6 70 0 1 2 3 0 1 2 31 1 2 0 4 1 2 0 42 2 0 1 5 3 0 1 23 3 4 5 6 2 7 4 5m\n 0 1 2 3 4 5 6 70 0 1 2 3 0 1 2 31 1 2 0 4 1 2 0 42 2 0 1 5 3 0 1 23 3 4 5 6 2 7 4 6One can also verify that S ∼ S′, but gS(3, 7) = 5, gS′(3, 7) = 6 for this case.References[ 1 ] C. L. Bouton, Nim, A Game with a Complete Mathematical Theory,Ann. of Math., 3 (1901–1902), 35–39.[ 2 ] A. Dress, A. Flammenkamp and N. Pink, Additive Periodicity of TheSprague-Grundy Function of Certain Nim Games, Adv. Appl. Math. 22(1999), 249–270.[ 3 ] S. Katayama and T. Kubo, Wythoﬀ’s Stone up Game and the Theoremof Rayleigh, Journal of the Tokushima Society for the History of Science37 (2018), to appear.[ 4 ] W. A. Liu, H. Li and B. Li, A Restricted Version of Wythoﬀ’s Nim,The Electric Journal of Combinatrics 18 (2011), # p 207.[ 5 ] F. Sato, On the Mathematics of Extracting Games of Stones-WonderfulRelations between Games and Algebra, Sugaku-Shobou, 2014 (in Japanese)[ 6 ] W. A. Wythoﬀ, A Modiﬁcation of the Game of Nim, Nieuw Arch, Wisk.7 (1907), 199–202.5

On Restricted Wythoff’s Nim

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On Restricted Wythoff’s Nim

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