Robert Voinescu, & Ryan Wood Department of Mathematics & Statistics, Northern Arizona University Deﬁnition Sylver Coinage is a game in which two players, A and B, alternately name positive integers that are not the sum of nonnegative multiples of previously named integers. The person that is forced to name 1 loses. Example Here is an example game between A and B. A: (5) = {5, 10, 15, . . .} B: (5, 4) = {4, 5, 8, 9, 10, 12, →} A: (5, 4, 11) = {4, 5, 8, →} B: (5, 4, 11, 6) = {4, 5, 6, 8, →} A: (5, 4, 11, 6, 7) = {4, →} B: (5, 4, 11, 6, 7, 2) = {2, 4, →} A: (5, 4, 11, 6, 7, 2, 3) = {2, →} B: Forced to choose 1 and loses! Known Results 1. Choosing 2 or 3 are bad opening moves for A. In either case, B can choose the other to guarantee a win. Choosing 4, 6, 8, 9, and 12 are also bad opening moves for A. The sequences (2, 3), (4, 6), (6, 9), and (8, 12) are winning positions for B. 2. Hutching’s Theorem: If gcd(a, b) = 1 and {a, b} = {2, 3}, then (a, b) is winning position for A. 3. If p is prime with p ≥ 5, then (p) is a winning position for A. 4. If n is composite & not equal to 2a3b, then (n) is a winning position for B. Open Question (John Conway) If player A names 16, and both players play optimally thereafter, then who wins? Note that 16 is the smallest number not handled by the facts above. Simpliﬁed Sylver Coinage In Simpliﬁed Sylver Coinage, A and B alternately name positive integers from the set [n] := {2, . . . , n} that are not the sum of nonnegative multiples of previously named numbers among [n]. The player that eliminates the last remaining number is the winner. Example Suppose n = 10. Below is one possible sequence of moves. A: (4) = {4, 8} B: (4, 5) = {4, 5, 8, 9, 10} A: (4, 5, 6) = {4, 5, 6, 8, 9, 10} B: (4, 5, 6, 3) = {3, 4, 5, 6, 7, 8, 9, 10} A: (4, 5, 6, 3, 2) = [10], and so A wins! Terminology of Impartial Games 1. An impartial game is a ﬁnite set X of positions together with a starting position and a collection {Opt(P) ≤ x | P ∈ X}, where Opt(P) is the set of possible options for a position P. 2. Two players take turns replacing the current position P with one of the available options in Opt(P). The player that encounters the empty option loses. 3. The minimum excludant mex(N) of a set of ordinals N is the smallest ordinal not contained in the set. Note that mex(∅) = 0. 4. The nim-number nim(P) of a position P is the mex of the set of nim- numbers of the options of P: nim(P) = mex{nim(Q)|Q ∈ Opt(P)}. 5. Terminal positions have nim-number 0. 6. Player B has a winning strategy iﬀ the nim-number for the game is 0. Example Game Trees If (x1, x2, . . . , xn ) is a sequence of moves, then the corresponding gap set is deﬁned via G(x1, x2, . . . , xn ) = [n] \ (x1, x2, . . . , xn ). Game tree for n = 2 [2]={2} mex(0)=1 (A wins) G(2)=∅ mex(∅)=0 2 Game tree for n = 3 [3]={2,3} mex(1,1)=0 (B wins) G(2)={3} mex(0)=1 G(3)={2} mex(0)=1 G(2,3)=∅ mex(∅)=0 G(3,2)=∅ mex(∅)=0 3 2 3 2 Game tree for n = 4 [4]={2,3,4} mex(1,2,0)=3 (A wins) G(3)={2,4} mex(0,1)=2 G(2)={3} mex(0)=1 G(4)={2,3} mex(1,1)=0 G(3,4)={2} mex(0)=1 G(3,2)=∅ mex(∅)=0 G(2,3)=∅ mex(∅)=0 G(4,2)={3} mex(0)=1 G(4,3)={2} mex(0)=1 G(3,4,2)=∅ mex(∅)=0 G(4,2,3)=∅ mex(∅)=0 G(4,3,2)=∅ mex(∅)=0 4 2 3 3 3 2 4 2 2 3 2 Data Table n Winner nim Strategy 2 A 1 A chooses 2 3 B 0 If A chooses 2 (resp 3), then B chooses 3 (resp 2) 4 A 3 A chooses 4 then mimics B’s strategy on n = 3 5 B 0 B stalls until A is forced to choose 2 or 3 6 A 3 A chooses 6 then mimics B’s strategy on n = 5 7 B 0 B stalls until A is forced to choose 2 or 3 8 A 5 A chooses 8 then mimics B’s strategy on n = 7 9 B 0 B intelligently stalls until A is forced to choose 2 or 3 10 A 1 A chooses 10 then mimics B’s strategy on n = 9 11 B 0 Complicated . . . 12 A 4 A chooses 12 then mimics B’s strategy on n = 11 Proposition If B wins on [n], then A wins on [n + 1]. Proof A’s opening move is n + 1. Then mimic B’s strategy on [n]. Corollary We never have two consecutive B’s if we continue our table. Question Will we ever encounter two consecutive A’s if we continue our table? Conjecture A wins on [n] iﬀ n is even. Potential strategy: A chooses n. Even if this is an appropriate strategy, there may be other winning strategies. One potential strategy to proving the conjecture is to show that if nim([n]) = 0, then nim([n + 2]) = 0. Knock Out In Knock Out (inspired by Simpliﬁed Sylver Coinage), A and B alternately name positive integers from the set [n], but this time we leave unselected numbers in play. The loser is the player that knocks out n as the sum of nonnegative multiples of previously named numbers among [n]. What happens in this game??? Example Suppose n = 6. Player A will lose immediately if they choose 2, 3, or 6. Suppose A chooses 4. Then B cannot choose 4 and again, 2, 3, or 6 are losing moves. If B is wise, they will choose 5. This forces A to choose 2, 3, or 6, and hence A loses. Email: Joni Hazelman [

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