A Family of Nim-Like Arrays: The Locator Theorem

Transcription

1 locator theorem paper v 2.pdf A Family of Nim-Like Arrays: The Locator Theorem Lowell Abrams a,denas.cowen-morton b, a Department of Mathematics, The George Washington University, Washington, DC USA b Department of Mathematics, Xavier University, Cincinnati, OH USA Abstract We study a property shared by some members of a certain family of Nim-like arrays. The arrays are recursively generated in a manner related to the operation of Nim-addition. The property involves a kind of self-referential relationship between entries and their indices. Mathematics Subject Classification: 91A46, 68R05 Keywords: Nim, sequential compound 1. Introduction The game of Nim is a two person combinatorial game consisting of at least one pile of stones in which the players alternate turns removing any number of stones they wish from asinglepileofstones;thewinneristheplayertotakethelast stone. Given combinatorial games G 1 and G 2,wecanformanewgamebytakingtheirdirectsumG 1 G 2 ;thisis the game in which a player, on their turn, has the option of making a move in exactly one of the games G 1 or G 2 as long as that game is not yet exhausted (in Nim this simply means having several independent piles of stones). Again, the winner is the last player to make a move. We call two games G 1 and G 2 equivalent if for every game H the game G 1 H has the same winner as the game G 2 H. Nim s importance was established by the Sprague-Grundy Theorem [5, 6] (also developed in [4, chapter 11]), which essentially asserts that Nim is universal among finite impartial two player combinatorial games in which the winner is the player to move last. Briefly, that is to say that every such game G is, vis-á-vis direct summation, equivalent to a single-pile Nim game. We denote by Nim s the Nim game with one pile of s stones. Corresponding author (Phone: Fax: ) addresses: labrams@gwu.edu (Lowell Abrams), morton@xavier.edu (Dena S. Cowen-Morton) Preprint submitted to Elsevier January 20, 2013

2 In [7], Stromquist and Ullman define sequential compounding, an operation on combinatorial games. In essence, the sequential compound G H of games G and H is the game in which play proceeds in G until there are no moves left, at which point play switches to H. Thus,ifH =Nim 0,wehaveG H = G. IfH =Nim 1,thenG H is misère play on G (in which the last player to take a stone loses rather than wins). Stromquist and Ullman consider games of the form (G 1 G 2 ) H, whereg 1,G 2, and H are independent impartial combinatorial games, but before our previous papers [1, 2], little was understood about this type of sequential compounding for H equivalent to Nim s with s>1. Such games give rise to a family A = {A s } s N0 of recursively generated arrays, and this family is the object of our study. Our (previous) articles [1, 2] explore some of the algebraic, combinatorial, and graphical properties of this family. In particular, [1] discusses how each A s may be viewed as defining on N 0 the structure of a loop, i.e., aquasigroupwithatwo-sidedidentity,and [2] presents periodicity properties which hold in the rows and [super]-diagonals of each A s. The results here continue our exploration of the family A,concentratingonourLocator Theorem, which establishes a connection between the row and column indices and the value of an entry. Letting a i,j denote the entry in row i, columnj of the array A s,thelocator Property states that the entry j in row i can be found in column a i,j.forarraysa s where s {0, 1, 2, 3, 5, 7} this Locator Property holds for all but a few specific pairs (i, j), but this seems to not be the case for any other value of s. Toquantifythis,weconcludewitha computational result studying the failure of the Locator Property in A s for all remaining values of s between 0 and 200. In particular, for all s {4, 6, 8, 9, 10,...,200}, thelocator Property fails for at least 70% of all pairs, and frequently more often. 2. Background: The Arrays A s We begin by constructing a family of infinite arrays using the mex operation: Definition 2.1. For a set X N 0 = {0, 1, 2, 3,...} we define mex(x) to be the smallest non-negative integer not contained in X, i.e., mex(x) =min(n 0 \ X). Here, mex stands for minimal excluded value. Definition 2.2. For any 2-dimensional array A indexed by N 0 N 0,leta i,j denote the entry in row i, columnj, wherei, j 0. The principal (p, q) subarray A(p, q) is the subarray of A consisting of entries a i,j with indices (i, j) {0,...,p} {0,...,q}. For i, j 0defineLeft(i, j)= {a i,q : q<j} to be the set of all entries in row i to the left of the entry a i,j,anddefineup(i, j)= {a p,j : p<i} to be the set of entries in column j above a i,j. 2

3 Observe that Definition 2.2 gives Left(i, 0) = Up(0,j)=. Definition 2.3. The infinite array A s,fors N 0,isconstructedrecursively: Theseed a 0,0 is set to s and for (i, j) (0, 0), ( ) a i,j := mex Left(i, j) Up(i, j). See, for example, Figures 1 and 2, which show principal subarrays in seeds 0, 2, and 3. In all figures the index i =0, 1, 2,... increases down the page and j =0, 1, 2,... increases to the right. These arrays will be useful in the calculations to come Figure 1: A 0(7, 7) Figure 2: A 2(7, 7) and A 3(7, 7) Note that both the top row (row i =0)andtheleftcolumn(columnj =0)ofA s always have entries in the order s, 0, 1, 2,...,s 1,s+1,s+2,... (see Property 3.7 below). The array A 0 is well known as the Nim addition table, and has been extensively studied in the setting of combinatorial game theory. We denote by Nim s the s-stone, single-pile Nim game and we denote by H the Grundy value of a game H, thatis,theuniquenumberto which H is equivalent (see [3] for details). Let G 1,G 2 be games with G 1 = i and G 2 = j (so G 1,G 2 are equivalent to Nim i, Nim j respectively). Then the (i, j)-entry of A 0 is equal 3

4 to the Grundy value G 1 G 2.Considerationofwhatisknownas misèreplay resultsin the array A 1. Using the sequential compounding construction of Stromquist and Ullman gives rise to the full family of arrays A s.indeed,the(i, j)-entry of A s is (G 1 G 2 ) Nim s. The arrays A s can be used to guide the actual play of a game (G 1 G 2 ) Nim s ;wedirect the reader to [1, 2] for more information. An important property of A s follow as immediate consequence of the recursive construction (see [1] for details): Proposition 2.4 ([1, Proposition 2.4]). For each s, thearraya s is symmetric, and each nonnegative integer appears exactly once in each row (and, by symmetry, each column). We conclude this section by defining (as in [2]) two different recursive algorithms which may be used to compute A s : Definition 2.5. By algorithm 1, we mean the algorithm described above in Definition 2.3, using the mex operation to fill in increasingly large subarrays containing the seed. Definition 2.6. By algorithm 2, which is well-defined only for principal subarrays A s (p, q), we first fill in all the 0 s, then all the 1 s, then all the 2 s, etc. To do so, we begin with just the seed s placed in the upper left hand corner. We start with k =0andsupposethatallentrieslessthank have been placed (if k =0thennothing other than the seed has been placed). Starting with row i =0,letm =min{j : k Up(i, j) and(i, j) isnotyetassignedanentry}; ifm q then set the (i, m) entrytok. Now increment i and, if i p, repeatthecalculationofm. Carry out this process for all k, incrementingk each time, until all entries in A s (p, q) havebeenfilled. Algorithm 2 succeeds in correctly filling out a finite portion of A s because when computing mex(x) forasetx, only those entries less than mex(x) areactuallyrelevanttothe calculation. 3. The Locator Theorem In this section we highlight an interesting property of the arrays A s that only holds for a small selection of seeds. We first noted this property in Section 7 of our paper [1]. Definition 3.1. For a fixed seed s we say that the Locator Property holds at (i, j) if a i,ai,j = j. Inwords,entryj in row i appears in column a i,j. Proposition 3.2. For both seeds 0 and 1, the Locator Property holds for all (i, j). Proof. As seed 0 represents two piles of stones being played in usual Nim, a i,j is found by bit-wise XOR [3]. Specifically, we can compute a i,j = i j. Sincebit-wiseXORisassociative,we 4

5 have a i,ai,j = a i,i j = i (i j) =(i i) j =0 j = j for all i, j, sothelocatorproperty holds for seed 0. Note now that A 1 differs from A 0 only in A 1 (1, 1). Thus, in seed 1, as long as either i>1 or j>1thesamecalculationasforseed0givesa i,ai,j = j. It is simple to confirm that each of the remaining four entries in A 1 (1, 1) also satisfies the Locator Property. Definition 3.3. A set X of column-indices is i-left-complete if the entries in row i equal to the elements of X appear in the columns indexed by a i,x = {a i,j : j X}. Definition 3.4. A set Y of row-indices is j-up-complete if the entries equal to j in the rows indexed by Y appear in the columns indexed by a Y,j = {a i,j : i Y }. We note two important properties of Up-complete and Left-complete sets that readily follow from the definitions: Property 3.5. Aunionofi-Left-complete sets is i-left-complete and a union of j-upcomplete sets is j-up-complete. Property 3.6. If the Locator Property holds at (i, j), thenx = {j} is i-left-complete and Y = {i} is j-up-complete. Next, we will need the following Properties from [1]. Property 3.8 may be proved using algorithm 2. Property 3.7. For all seeds s and all k, s if k =0 a 0,k = a k,0 = k 1 if 0 <k s k if k>s. Property 3.8. For all seeds s 1 and all k, Lemma 3.9. For all seeds s 2, s if k =0 a k,k = 1 if k =1 0 otherwise. 1. The set {0, 1,...,s} is 0-Left-complete and 0-Up-complete, but {0, 1,...,k} for k<s is neither 0-Left-complete nor 0-Up-complete. 2. If {0, 1,...,j 1} is 0-Left-complete, then the Locator Property holds at (0,j). 3. If {0, 1,...,i 1} is 0-Up-complete, then the Locator Property holds at (i, 0). 5

6 Proof. To demonstrate the first assertion we use Properties 3.7 and 3.8 repeatedly. That {0, 1,...,k} for k<sis not 0-Left-complete when s 2followsbynotingthatifk<sthen in row 0, the entry k 1occursincolumnk a 0,X = {a 0,0,...,a 0,k } = {s, 0, 1,...,k 1}. However, if X = {0, 1,...,s} then a 0,X = X, sowedohave0-left-completeness. Let Y = {0, 1,...,k}. For k =0wehavea Y,0 = {s}, butsinces>1anda 0,1 =0,by Property 2.4, 0-Up-completeness fails. By Property 3.8, for k>0theentriesequalto0in rows indexed by Y occur in the columns indexed by Y.SincewehaveY a Y,0 for k<s and Y = a Y,0 for k = s, 0-Up-completenessholdsfork = s but not for k<s. By the hypotheses of the second and third assertions, the first assertion implies that i, j s. By Property 3.7 we have a 0,j = j and a i,0 = i, andthesecondandthirdassertionsreadily follow from Properties 3.7 and 3.8. Lemma For i, j 1, if{0, 1,...,(j 1)} is i-left-complete and {0, 1,...,(i 1)}) is j-up-complete, then the Locator Property holds at (i, j). Proof. Since {0, 1,...,(i 1)} is j-up-complete, the entries in Up(i, j) ={a 0,j,...,a (i 1),j } tell us the columns in which entry j appears for rows 0 through i 1. Since j is already in each of these columns, Proposition 2.4 tells us that in row i the entry j is not in any of the columns enumerated in {a 0,j,...,a (i 1),j }.Likewise,since{0, 1,...,(j 1)} is i-leftcomplete, looking at Left(i, j) ={a i,0,...,a i,(j 1) },wefindthecolumnnumbersinwhich j cannot appear in row i since there are already other entries in these columns, namely, the entries 0,...,(j 1). We now have a list of columns in which j does not appear in row i; thosearisingbecause j is already in that column (enumerated in Up(i, j) inarowofsmallerindex),andthose arising because an entry smaller than j is already in that column (enumerated in Left(i, j)) in row i. If we are using algorithm 2 for filling in the mex table, so that all values appear as early as they can, then the entry j appears in the first non-forbidden column in row i i.e., in the column with the smallest non-forbidden index. But this ispreciselythemexof the forbidden ( column numbers. ) Thus the entry j in row i goes into the row numbered m =mex Left(i, j) Up(i, j) = a i,j.inotherwords,entryj in row i appears in column a i,j,soa i,ai,j = j. Lemma For seed s =2,theset{0, 1, 2} is both i-left-complete for i =0, 1, 2 and j-up-complete for j =0, 1, 2. Lemma For seed s =3,thefollowinghold: 1. The set {0, 1, 2, 3} is i-left-complete for i =0, 1, 2 and 0-Up-complete. 2. The set {0, 1, 2, 3, 4} is 3-Up-complete and 3-Left-complete. 6

7 3. The set {1, 2, 3, 4, 5} is 4-Left-complete. 4. The sets {1, 2} and {i 1,i,i+1} are i-left-complete for all i The set {j 1,j,j +1} is j-up-complete for all j 4. These lemmas can readily be proven by induction using Figure 2; the details are left to the reader, but note that the patterns are established by considering even and odd cases and that [2] provides many useful lemmas. We do not include the necessary lemmas for seeds 5and7,butthereadercaneasilysurmisethem. Theorem 3.13 (The Locator Theorem). For seeds s =2, 3, 5, 7, thelocatorpropertyholdsforall(i, j) L s where L 2 = {(i, j) :i>sor j>s}, L 3 = {(i, j) :(i>sor j>s) and j 1, 2 and i j > 1}, L 5 = {(i, j) :(i>s+1 or j>s+1) and j 1, 2, 3, 4 and i j > 2}, L 7 = {(i, j) :(i>s+2 or j>s+2) and j 1, 2, 3, 4, 5 and i j > 3}. Proof. We use Lemmas 3.9, 3.11, and 3.12, as well as their analogues for seeds 5 and 7, as base cases. Suppose (i, j) L s for s {2, 3, 5, 7}, andmoreoverthatthelocator Property holds for all (i,j ) L s with i <ior j <j.bythebasecasesandproperty3.6 it follows that {0, 1,...,(j 1)} is a union of one or more sets which are i-left-complete and {0, 1,...,(i 1)} is a union of one or more sets which are j-up-complete. By Property 3.5, {0, 1,...,(j 1)} is i-left-complete and {0, 1,...,(i 1)} is j-up-complete. Lemma 3.10 therefore proves that the Locator Property holds at (i, j). 4. A Computational Result We conclude by computationally examining, for each seed from 0 to 200, the percentage of entries for which the Locator Property holds (this is expressed more precisely below). Our results suggest that the Locator Property does not hold for seeds other than s = 0, 1, 2, 3, 5, 7. Theorem 4.1. In A s,let P s = { (i, j) {0, 1, 2,...,2999} 2 : j>s and s<a i,j < 3000 }, and ρ s = Q s / P s.wehave Q s = {(i, j) P s : a i,ai,j = j}, if s {0, 1, 2, 3, 5, 7} then ρ s =1; if s {4, 6} {8, 9,...,200} then ρ s (0.155, 0.279). 7

8 ρ s seed Figure 3: The values ρ s for seeds s {4, 6, 8, 9, 10,...}. Thus ρ s gives a measure of the extent to which entries in A s satisfy the Locator Property. Figure 3 displays the values of ρ s for seeds other than {0, 1, 2, 3, 5, 7}. Note that in the definition of P s the conditions j>sand a i,j >sreflect the fact that for entries with small column indices and entries near the diagonal, respectively, we only have Left-completeness for non-singleton sets (see Lemmas 3.11 and 3.12). 5. Acknowledgements Thomas Zaslavsky offered helpful suggestions during the writing stage. 6. References [1] L. Abrams, D. Cowen-Morton, Algebraic Structure in a Family of Nim-like Arrays, J. Pure and Applied Algebra Vol. 214 (2010) [2] L. Abrams, D. Cowen-Morton, Periodicity and Other Structure in a Colorful Family of Nim-like Arrays, Electronic Journal of Combinatorics Vol. 17 (2010). [3] E. R. Berlekamp, H. H. Conway, R. K. Guy, Winning Ways For Your Mathematical Plays Vol. 1, secondedition,a.k.peters,wellesley,massachusetts,2001. [4] J. H. Conway, On Numbers and Games, secondedition,a.k.peters,wellesley,massachusetts, [5] P. M. Grundy, Mathematics and Games, Eureka, Vol. 2(1939)

9 [6] R. P. Sprague, Über mathematische Kampfspiele. Tôhoku Math. J. Vol. 41 (1935-6) [7] W. Stromquist, D. Ullman, Sequential compounds of combinatorial games, Theoret. Computer Science Vol. 119 (1993)