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A family of Nim-like arrays: The Locator Theorem
Theoretical Computer Science 535 (2014) 31–37
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Theoretical Computer Science www.elsevier.com/locate/tcs
A family of Nim-like arrays: The Locator Theorem Lowell Abrams a , Dena S. Cowen-Morton b,∗ a b
Department of Mathematics, The George Washington University, Washington, DC 20052, USA Department of Mathematics, Xavier University, Cincinnati, OH 45207-4441, USA
a r t i c l e
i n f o
Article history: Received 21 January 2013 Received in revised form 29 September 2013 Accepted 27 March 2014 Communicated by A. Fraenkel
a b s t r a c t 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, and the property involves a kind of self-referential relationship between entries and their indices. We conclude with some numerical evidence that the property in fact holds for only a few members of the family. © 2014 Published by Elsevier B.V.
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 a single pile of stones; the winner is the player to take the last stone. Given combinatorial games G 1 and G 2 , we can form a new game by taking their direct sum G 1 ⊕ G 2 ; this is 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 [7,8] (also developed in [6, 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-à-vis direct summation, equivalent to a single-pile Nim game. We denote by Nims the Nim game with one pile of s stones. In [9], 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, if H = Nim0 , we have G → H = G. If H = Nim1 , then G → 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 , where G 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 Nims with s > 1. Such games give rise to a family A∗ = {As }s∈N0 of recursively generated arrays (we write N0 for N ∪ {0}), and this family is the object of our study. Here, s is the Grundy-value (see [4]) of H , and for a fixed value of s the rows and columns of As are indexed by the Grundy values of G 1 and G 2 , respectively. Our papers [1,2] explore some of the algebraic, combinatorial, and graphical properties of this family. In particular, [1] discusses how each As may be viewed as defining on N0 the structure
*
Corresponding author. Tel.: +1 513 745 3674; fax: +1 513 745 3272. E-mail addresses: [email protected] (L. Abrams), [email protected] (D.S. Cowen-Morton).
http://dx.doi.org/10.1016/j.tcs.2014.03.032 0304-3975/© 2014 Published by Elsevier B.V.
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L. Abrams, D.S. Cowen-Morton / Theoretical Computer Science 535 (2014) 31–37
Fig. 1. A Nim-like game.
of a loop, i.e., a quasigroup with a two-sided identity, and [2] presents periodicity properties which hold in the rows and [super]-diagonals of each As . The results here continue our exploration of the family A∗ , concentrating on our “Locator Theorem,” which establishes a connection between the row and column indices and the value of an entry. More explicitly, letting ai , j denote the entry in row i, column j of the array As and defining the operator mi : N0 → N0 by mi ( j ) := ai , j , we say the Locator Property holds at (i , j ) if there is a k ∈ N0 such that mi ( j ) = k and mi (k) = j. Alternatively, the Locator Property may be formulated as asserting that, in row i, the entry equal to j is located in column ai , j . For arrays As 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. To quantify this, we conclude with a computational result studying the failure of the Locator Property in As for all remaining values of s between 0 and 200. In particular, for all s ∈ {4, 6, 8, 9, 10, . . . , 200}, the Locator Property fails for at least 70% of all pairs, and frequently more often. Interestingly, the Locator Theorem gives information about a Nim-like game we now describe. Consider the arrangement of rectangles shown in Fig. 1, where each of i , j , s denotes a nonnegative integer, and suppose that each box contains the quantity of stones indicated by the corresponding variable. On each player’s turn they may choose one box and remove any number of stones from that box, subject to the condition that a box may only be chosen if all the boxes directly above it in the diagram are empty. The Locator Theorem tells us that if s ∈ {0, 1, 2, 3, 5, 7}, then for all but a few values of i , j (the restrictions depending on s) the game is a second-player win. Further details are given at the end of Section 3. Another interesting application of the Locator Theorem is developed in [3]. For each value of s one can associate to As a group of affine permutations of the integers [5]; the Locator Theorem tells us that for s ∈ {0, 1, 2, 3, 5, 7} these groups are generated by involutions. 2. Background: the arrays As We begin by constructing a family of infinite arrays using the mex operation: Definition 2.1. For a set X ⊆ N0 = {0, 1, 2, 3, . . .} we define mex( X ) to be the smallest non-negative integer not contained in X , i.e.,
mex( X ) = min(N0 \ X ). Here, mex stands for minimal excluded value. Definition 2.2. For any 2-dimensional array A indexed by N0 × N0 , let ai , j denote the entry in row i, column j, where i , j 0. The principal ( p , q) subarray A( p , q) is the subarray of A consisting of entries ai , j with indices (i , j ) ∈ {0, . . . , p } × {0, . . . , q}. For i , j 0 define Left(i , j ) = {ai ,q : q < j } to be the set of all entries in row i to the left of the entry ai , j , and define Up(i , j ) = {a p , j : p < i } to be the set of entries in column j above ai , j . Observe that Definition 2.2 gives Left(i , 0) = Up(0, j ) = ∅. Definition 2.3. The infinite array As , for s ∈ N0 , is constructed recursively: The seed a0,0 is set to s and for (i , j ) = (0, 0),
ai , j := mex Left(i , j ) ∪ Up(i , j ) . See, for example, Fig. 2, which shows principal subarrays in seeds 0, 2, and 3. In all arrays 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. Note that both the top row (row i = 0) and the left column (column j = 0) of As always have entries in the order s, 0, 1, 2, . . . , s − 1, s + 1, s + 2, . . . (see Property 2.8 below). The array A0 is well known as the Nim addition table, and has been extensively studied in the setting of combinatorial game theory. We denote by Nims the s-stone, single-pile Nim game and we denote by | H | the Grundy value of a game H , that is, the unique number to which H is equivalent (see [4] 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 Nimi , Nim j respectively). Then the (i , j )-entry of A0 is equal to the Grundy value |G 1 ⊕ G 2 |. Consideration of what is known as “misère play” results in the array A1 . Using the sequential compounding construction
L. Abrams, D.S. Cowen-Morton / Theoretical Computer Science 535 (2014) 31–37
⎡
0
1 0 3 2 5 4 7 6
⎢1 ⎢ ⎢2 ⎢ ⎢3 ⎢ ⎢4 ⎢ ⎢5 ⎢ ⎣6 7
⎡
2 0 ⎢0 1 ⎢ ⎢1 2 ⎢ ⎢3 4 ⎢ ⎢4 3 ⎢ ⎢5 6 ⎢ ⎣6 5 7 8
1 2 0 5 6 3 4 9
3 4 5 0 1 2 7 6
4 3 6 1 0 7 2 5
5 6 3 2 7 0 1 4
6 5 4 7 2 1 0 3
2 3 0 1 6 7 4 5
⎤
3 2 1 0 7 6 5 4
7 8⎥ ⎥ 9⎥ ⎥ 6⎥ ⎥ 5⎥ ⎥ 4⎥ ⎥ 3⎦ 0
4 5 6 7 0 1 2 3
⎡
5 4 7 6 1 0 3 2
6 7 4 5 2 3 0 1
3
0 1 2 3 5 4 7 7 6
⎤
7 6⎥ ⎥ 5⎥ ⎥ 4⎥ ⎥ 3⎥ ⎥ 2⎥ ⎥ 1⎦ 0 1 2 0 4 3 6 5 8
⎢0 ⎢ ⎢1 ⎢ ⎢2 ⎢ ⎢4 ⎢ ⎢5 ⎢ ⎣6
33
2 4 3 5 4 3 0 1 1 0 7 2 8 9 5 10
5 4 6 7 2 0 1 3
6 7 5 8 9 1 0 2
⎤
7 6 ⎥ ⎥ 8 ⎥ ⎥ 5 ⎥ ⎥ 10 ⎥ ⎥ 3 ⎥ ⎥ 2 ⎦ 0
Fig. 2. A0 (7, 7) (above) and A2 (7, 7) and A3 (7, 7) (below, left and right).
⎡
0 ⎢1 ⎢ ⎢2 ⎢ ⎢3 ⎢ ⎢4 ⎢ ⎢5 ⎢ ⎣6 7
1 2 0 3 3 0 2 1 5 6 4 7 7 4 6 5
3 2 1 0 7 6 5 4
4 5 6 7 0 1 2 3
5 4 7 6 1 0 3 2
6 7 4 5 2 3 0 1
7 6 5 4 3 2 1 0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎡
2 ⎢0 ⎢ ⎢1 ⎢ ⎢3 ⎢ ⎢4 ⎢ ⎢5 ⎢ ⎣6 7
0 1 2 4 3 6 5 8
1 2 0 5 6 3 4 9
3 4 5 0 1 2 7 6
4 3 6 1 0 7 2 5
5 6 3 2 7 0 1 4
6 5 4 7 2 1 0 3
7 8 9 6 5 4 3 0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Fig. 3. Some examples of the Locator Theorem. In each array, the elements involved are boxed. Note that the relevant row and column indices are, for these examples, the corresponding entries in column 0 and row 0, respectively.
of Stromquist and Ullman gives rise to the full family of arrays As . Indeed, the (i , j )-entry of As is |(G 1 ⊕ G 2 ) → Nims |. The arrays As can be used to guide the actual play of a game (G 1 ⊕ G 2 ) → Nims ; we direct the reader to [1, §2] for more information. A fundamental property of As follows as an immediate consequence of the recursive construction (see [1] for details): Proposition 2.4. (See [1], Proposition 2.4.) For each s, the array As is symmetric, and each nonnegative integer appears exactly once in each row (and, by symmetry, each column). From this proposition, it easily follows that for each i ∈ N0 , the function mi : N0 → N0 defined by mi ( j ) := ai , j is a bijection. The main focus of this paper is the exploration of the following “Locator Property” in the arrays As for a variety of seeds s: Definition 2.5. For a fixed seed s we say that the Locator Property holds at (i , j ) if there is an integer k such that mi ( j ) = k and mi (k) = j, i.e., if, in the cycle decomposition of mi , the element j appears in a transposition. Equivalently, the Locator Property holds at (i , j ) if we have ai ,ai, j = j, i.e., in row i the entry equal to j is located in column ai , j . Several related illustrations of the Locator Theorem are highlighted in each array in Fig. 3. For seed s = 0 (in the lefthand array) we have, for example, a3,4 = m3 (4) = 7 and a3,7 = m3 (m3 (4)) = 4. Similarly, a4,7 = m4 (7) = 3 and a4,3 = m4 (m4 (7)) = 7. Turning now to seed s = 2 (in the right-hand array) we have a4,5 = m4 (5) = 7 and a4,7 = m4 (m4 (5)) = 5, as well as a7,4 = m7 (4) = 5 and a7,5 = m7 (m7 (4)) = 4. The reader can easily discern additional examples among the highlighted entries. We end this section with a recapitulation of some key definitions and propositions from our papers [1,2]. Definition 2.6. 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 2.7. By algorithm 2, which is well-defined only for principal subarrays As ( 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 = 0 and suppose that all entries less than k have been placed (if k = 0 then nothing other than the seed has been / Up(i , j ) and (i , j ) is not yet assigned an entry}; if m q then set the placed). Starting with row i = 0, let m = min{ j: k ∈
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L. Abrams, D.S. Cowen-Morton / Theoretical Computer Science 535 (2014) 31–37
(i , m) entry to k. Now increment i and, if i p, repeat the calculation of m. Carry out this process for all k, incrementing k each time, until all entries in As ( p , q) have been filled. Algorithm 2 succeeds in correctly filling out a finite portion of As because when computing mex( X ) for a set X , only those entries less than mex( X ) are actually relevant to the calculation. As proved in [2], the rows, columns, and diagonals in As do eventually settle into distinct periodic or arithmetically periodic patterns for s 2. These patterns are frequently complicated to compute, with the behavior in the first few rows and diagonals the easiest to find by hand. As an example, we give the following properties from [1], both of which will be needed below. For the reader who wishes to consult reference [1], which makes use of an operation ∗, we note that for small values of i , j we have mi ( j ) = i ∗ j, but for i , j > s we do have mi ( j ) = i ∗ j. Property 2.8. For all seeds s and all k,
a0,k = ak,0 =
⎧ ⎨s
k−1 ⎩ k
if k = 0 if 0 < k s if k > s.
Property 2.9. For all seeds s 1 and all k,
ak,k =
⎧ ⎨s 1 ⎩ 0
if k = 0 if k = 1 otherwise.
Property 2.9 may easily be verified using algorithm 2. 3. The Locator Theorem We first discuss seeds 0 and 1 since these behave very differently from other seeds. Proposition 3.1. 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, ai , j is found by bit-wise XOR [4]. Specifically, we can compute ai , j = i ⊕ j. Since bit-wise XOR is associative, we have mi (mi ( j )) = mi (i ⊕ j ) = i ⊕ (i ⊕ j ) = (i ⊕ i ) ⊕ j = 0 ⊕ j = j for all i , j, so the Locator Property holds for seed 0. Note now that A1 differs from A0 only in A1 (1, 1). Thus, in seed 1, as long as either i > 1 or j > 1 the same calculation as for seed 0 gives mi (mi ( j )) = j. It is simple to confirm that each of the remaining four entries in A1 (1, 1) also satisfies the Locator Property. 2 The proof of the remainder of our main result rests on the idea of a set of indices being Left- or Up-complete, which we define here. Even though they are similar, Left- and Up-completeness do not work quite the same. Definition 3.2. A set X of column-indices is i-Left-complete if, in row i, the entries equal to the elements of X are located in the columns indexed by mi ( X ) = {mi ( j ): j ∈ X }. A set Y of row-indices is j-Up-complete if, in the rows indexed by Y , the entries equal to j are located in the columns indexed by m Y ( j ) = {mi ( j ): i ∈ Y }. Before proceeding with an example, we remind the reader that rows and columns are indexed from 0. For example, in seed s = 3 (see Fig. 4), the set {0, 1, 2, 3} is 2-Left-complete because the entries in row 2 equal to 0, 1, 2, 3 are located in columns indexed by 2, 0, 1, 4 (respectively), and m2 ({0, 1, 2, 3}) = {1, 2, 0, 4}. Similarly, the set Y = {0, 1, 2, 3, 4} is 3-Up-complete in seed s = 3 because the entries equal to j = 3 in rows 0, 1, 2, 3, 4 are located in columns 0, 3, 4, 1, 2 (respectively) and m{0,1,2,3,4} (2) = {1, 2, 0, 4, 3}. Note that for Left-completeness, we are looking for the column locations of several values in a single row, whereas in Up-completeness, we are looking for the column locations of a single value in several rows. We note two important properties of Up-complete and Left-complete sets that readily follow from the definitions: Property 3.3. A union of i-Left-complete sets is i-Left-complete and a union of j-Up-complete sets is j-Up-complete. Property 3.4. If the Locator Property holds at (i , j ), then X = { j } is i-Left-complete and Y = {i } is j-Up-complete.
L. Abrams, D.S. Cowen-Morton / Theoretical Computer Science 535 (2014) 31–37
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Fig. 4. Illustrations in seed 3 of 2-Left-completeness (on the left) and 3-Up-completeness (on the right). On the left, the circled entries in row 2 are located in the columns whose indices are in the horizontal rectangle in row 2. On the right, the entries equal to 3 are in the columns whose indices are in the vertical rectangle in column 3.
Lemma 3.5. For all seeds s 2, 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-Upcomplete. 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). Proof. To demonstrate the first assertion we use Properties 2.8 and 2.9 repeatedly. That {0, 1, . . . , k} for k < s is not 0-Left-complete when s 2 follows by noting that if k < s then in row 0, the entry k − 1 occurs in column k ∈ / m0 ( X ) = {m0 (0), . . . , m0 (k)} = {s, 0, 1, . . . , k − 1}. However, if X = {0, 1, . . . , s} then m0 ( X ) = X , so we do have 0-Left-completeness. Let Y = {0, 1, . . . , k}. For k = 0 we have m Y (0) = {s}, but since s > 1 and a0,1 = 0, by Property 2.4, 0-Up-completeness fails. By Property 2.9, for k > 0 the entries equal to 0 in rows indexed by Y occur in the columns indexed by Y . Since we have Y = m Y (0) for k < s and Y = m Y (0) for k = s, 0-Up-completeness holds for k = 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 2.8 we have a0, j = j and ai ,0 = i. The second assertion readily follows. By Property 2.9, the third assertion also follows. 2 Lemma 3.6. 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 ) = {a0, 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 {a0, j , . . . , a(i −1), j }. Likewise, since {0, 1, . . . , ( j − 1)} is i-Left-complete, looking at Left(i , j ) = {ai ,0 , . . . , ai ,( j −1) }, we find the column numbers in which 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; those arising because j is already in that column (enumerated in Up(i , j ) in a row of smaller index), and those 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 is precisely the mex of the forbidden column numbers. Thus the entry j in row i goes into the column numbered m = mex(Left(i , j ) ∪ Up(i , j )) = ai , j . In other words, entry j in row i appears in column ai , j , so ai ,ai, j = j, i.e., mi (mi ( j )) = j. 2 Lemma 3.7. For seed s = 2, the set {0, 1, 2} is both i-Left-complete for i = 0, 1, 2 and j-Up-complete for j = 0, 1, 2. Lemma 3.8. For seed s = 3, the following hold: 1. 2. 3. 4. 5.
The set {0, 1, 2, 3} is i-Left-complete for i = 0, 1, 2 and 0-Up-complete. The set {0, 1, 2, 3, 4} is 3-Up-complete and 3-Left-complete. The set {1, 2, 3, 4, 5} is 4-Left-complete. The sets {1, 2} and {i − 1, i , i + 1} are i-Left-complete for all i 5. The set { j − 1, j , j + 1} is j-Up-complete for all j 4.
These lemmas can readily be proven by induction using Fig. 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 5 and 7, but the reader can easily surmise them.
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L. Abrams, D.S. Cowen-Morton / Theoretical Computer Science 535 (2014) 31–37
Fig. 5. The values
ρs for seeds s ∈ {4, 6, 8, 9, 10, . . .}.
Theorem 3.9 (The Locator Theorem). For seeds s = 2, 3, 5, 7, the Locator Property holds for all (i , j ) ∈ L s where L 2 = {(i , L 3 = {(i , L 5 = {(i , L 7 = {(i ,
j ): j ): j ): j ):
i > s or j > s}, (i > s or j > s) and j = 1, 2 and |i − j | > 1}, (i > s + 1 or j > s + 1) and j = 1, 2, 3, 4 and |i − j | > 2}, (i > s + 2 or j > s + 2) and j = 1, 2, 3, 4, 5 and |i − j | > 3}.
Proof. We use Lemmas 3.5, 3.7, and 3.8, as well as their analogues for seeds 5 and 7, as base cases. Suppose (i , j ) ∈ L s for s ∈ {2, 3, 5, 7}, and moreover that the Locator Property holds for all (i , j ) ∈ L s with i < i or j < j. By the base cases and Property 3.4 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 3.3, {0, 1, . . . , ( j − 1)} is i-Leftcomplete and {0, 1, . . . , (i − 1)} is j-Up-complete. Lemma 3.6 therefore proves that the Locator Property holds at (i , j ). 2 We now return to the game described in the introduction and depicted in Fig. 1. That game can be described as
(Nim j ⊕ Nimi ) → Nims ⊕ Nimi → Nims ⊕ Nim j , and the Grundy value of this game is exactly Nimmi (mi ( j )) ⊕ Nim j , where mi is defined in terms of the array As . Thus, the game is a second player win exactly when mi (mi ( j )) = j, i.e., when the Locator Property holds at (i , j ). The beginning of a winning strategy for the second player would be to mirror any move by the first player in an “i” pile in the other “i” pile, and similarly for the “ j” pile, as long as this returns the game to a configuration for which the Locator Property holds. 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 As , let
P s = (i , j ) ∈ {0, 1, 2, . . . , 2999}2 : j > s and s < ai , j < 3000 ,
Q s = (i , j ) ∈ P s : ai ,ai, j = j , and ρs = | Q s |/| P s |. We have
if s ∈ {0, 1, 2, 3, 5, 7} then ρs = 1; if s ∈ {4, 6} ∪ {8, 9, . . . , 200} then ρs ∈ (0.155, 0.279). Thus ρs gives a measure of the extent to which entries in As satisfy the Locator Property. Fig. 5 displays the values of ρs for seeds other than {0, 1, 2, 3, 5, 7}. Note that in the definition of P s the conditions j > s and ai , j > s reflect 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 3.7 and 3.8).
L. Abrams, D.S. Cowen-Morton / Theoretical Computer Science 535 (2014) 31–37
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Acknowledgements Thomas Zaslavsky offered helpful suggestions during the writing stage. In addition, comments from the anonymous referees improved the exposition significantly. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L. Abrams, D. Cowen-Morton, Algebraic structure in a family of Nim-like arrays, J. Pure Appl. Algebra 214 (2010) 165–176. L. Abrams, D. Cowen-Morton, Periodicity and other structure in a colorful family of Nim-like arrays, Electron. J. Combin. 17 (2010). L. Abrams, D. Cowen-Morton, A family of Nim-like arrays: stabilization, 2013, preprint. E.R. Berlekamp, H.H. Conway, R.K. Guy, Winning Ways for Your Mathematical Plays, vol. 1, second edition, AK Peters, Wellesley, Massachusetts, 2001. A. Björner, F. Brenti, Affine permutations of type A, Electron. J. Combin. 3 (1996). J.H. Conway, On Numbers and Games, second edition, AK Peters, Wellesley, Massachusetts, 2001. P.M. Grundy, Mathematics and games, Eureka 2 (1939) 6–8. R.P. Sprague, Über mathematische Kampfspiele, Tohoku Math. J. (2) 41 (1935–1936) 438–444. W. Stromquist, D. Ullman, Sequential compounds of combinatorial games, Theoret. Comput. Sci. 119 (1993) 311–321.
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Theoretical Computer Science www.elsevier.com/locate/tcs
A family of Nim-like arrays: The Locator Theorem Lowell Abrams a , Dena S. Cowen-Morton b,∗ a b
Department of Mathematics, The George Washington University, Washington, DC 20052, USA Department of Mathematics, Xavier University, Cincinnati, OH 45207-4441, USA
a r t i c l e
i n f o
Article history: Received 21 January 2013 Received in revised form 29 September 2013 Accepted 27 March 2014 Communicated by A. Fraenkel
a b s t r a c t 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, and the property involves a kind of self-referential relationship between entries and their indices. We conclude with some numerical evidence that the property in fact holds for only a few members of the family. © 2014 Published by Elsevier B.V.
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 a single pile of stones; the winner is the player to take the last stone. Given combinatorial games G 1 and G 2 , we can form a new game by taking their direct sum G 1 ⊕ G 2 ; this is 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 [7,8] (also developed in [6, 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-à-vis direct summation, equivalent to a single-pile Nim game. We denote by Nims the Nim game with one pile of s stones. In [9], 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, if H = Nim0 , we have G → H = G. If H = Nim1 , then G → 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 , where G 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 Nims with s > 1. Such games give rise to a family A∗ = {As }s∈N0 of recursively generated arrays (we write N0 for N ∪ {0}), and this family is the object of our study. Here, s is the Grundy-value (see [4]) of H , and for a fixed value of s the rows and columns of As are indexed by the Grundy values of G 1 and G 2 , respectively. Our papers [1,2] explore some of the algebraic, combinatorial, and graphical properties of this family. In particular, [1] discusses how each As may be viewed as defining on N0 the structure
*
Corresponding author. Tel.: +1 513 745 3674; fax: +1 513 745 3272. E-mail addresses: [email protected] (L. Abrams), [email protected] (D.S. Cowen-Morton).
http://dx.doi.org/10.1016/j.tcs.2014.03.032 0304-3975/© 2014 Published by Elsevier B.V.
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L. Abrams, D.S. Cowen-Morton / Theoretical Computer Science 535 (2014) 31–37
Fig. 1. A Nim-like game.
of a loop, i.e., a quasigroup with a two-sided identity, and [2] presents periodicity properties which hold in the rows and [super]-diagonals of each As . The results here continue our exploration of the family A∗ , concentrating on our “Locator Theorem,” which establishes a connection between the row and column indices and the value of an entry. More explicitly, letting ai , j denote the entry in row i, column j of the array As and defining the operator mi : N0 → N0 by mi ( j ) := ai , j , we say the Locator Property holds at (i , j ) if there is a k ∈ N0 such that mi ( j ) = k and mi (k) = j. Alternatively, the Locator Property may be formulated as asserting that, in row i, the entry equal to j is located in column ai , j . For arrays As 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. To quantify this, we conclude with a computational result studying the failure of the Locator Property in As for all remaining values of s between 0 and 200. In particular, for all s ∈ {4, 6, 8, 9, 10, . . . , 200}, the Locator Property fails for at least 70% of all pairs, and frequently more often. Interestingly, the Locator Theorem gives information about a Nim-like game we now describe. Consider the arrangement of rectangles shown in Fig. 1, where each of i , j , s denotes a nonnegative integer, and suppose that each box contains the quantity of stones indicated by the corresponding variable. On each player’s turn they may choose one box and remove any number of stones from that box, subject to the condition that a box may only be chosen if all the boxes directly above it in the diagram are empty. The Locator Theorem tells us that if s ∈ {0, 1, 2, 3, 5, 7}, then for all but a few values of i , j (the restrictions depending on s) the game is a second-player win. Further details are given at the end of Section 3. Another interesting application of the Locator Theorem is developed in [3]. For each value of s one can associate to As a group of affine permutations of the integers [5]; the Locator Theorem tells us that for s ∈ {0, 1, 2, 3, 5, 7} these groups are generated by involutions. 2. Background: the arrays As We begin by constructing a family of infinite arrays using the mex operation: Definition 2.1. For a set X ⊆ N0 = {0, 1, 2, 3, . . .} we define mex( X ) to be the smallest non-negative integer not contained in X , i.e.,
mex( X ) = min(N0 \ X ). Here, mex stands for minimal excluded value. Definition 2.2. For any 2-dimensional array A indexed by N0 × N0 , let ai , j denote the entry in row i, column j, where i , j 0. The principal ( p , q) subarray A( p , q) is the subarray of A consisting of entries ai , j with indices (i , j ) ∈ {0, . . . , p } × {0, . . . , q}. For i , j 0 define Left(i , j ) = {ai ,q : q < j } to be the set of all entries in row i to the left of the entry ai , j , and define Up(i , j ) = {a p , j : p < i } to be the set of entries in column j above ai , j . Observe that Definition 2.2 gives Left(i , 0) = Up(0, j ) = ∅. Definition 2.3. The infinite array As , for s ∈ N0 , is constructed recursively: The seed a0,0 is set to s and for (i , j ) = (0, 0),
ai , j := mex Left(i , j ) ∪ Up(i , j ) . See, for example, Fig. 2, which shows principal subarrays in seeds 0, 2, and 3. In all arrays 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. Note that both the top row (row i = 0) and the left column (column j = 0) of As always have entries in the order s, 0, 1, 2, . . . , s − 1, s + 1, s + 2, . . . (see Property 2.8 below). The array A0 is well known as the Nim addition table, and has been extensively studied in the setting of combinatorial game theory. We denote by Nims the s-stone, single-pile Nim game and we denote by | H | the Grundy value of a game H , that is, the unique number to which H is equivalent (see [4] 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 Nimi , Nim j respectively). Then the (i , j )-entry of A0 is equal to the Grundy value |G 1 ⊕ G 2 |. Consideration of what is known as “misère play” results in the array A1 . Using the sequential compounding construction
L. Abrams, D.S. Cowen-Morton / Theoretical Computer Science 535 (2014) 31–37
⎡
0
1 0 3 2 5 4 7 6
⎢1 ⎢ ⎢2 ⎢ ⎢3 ⎢ ⎢4 ⎢ ⎢5 ⎢ ⎣6 7
⎡
2 0 ⎢0 1 ⎢ ⎢1 2 ⎢ ⎢3 4 ⎢ ⎢4 3 ⎢ ⎢5 6 ⎢ ⎣6 5 7 8
1 2 0 5 6 3 4 9
3 4 5 0 1 2 7 6
4 3 6 1 0 7 2 5
5 6 3 2 7 0 1 4
6 5 4 7 2 1 0 3
2 3 0 1 6 7 4 5
⎤
3 2 1 0 7 6 5 4
7 8⎥ ⎥ 9⎥ ⎥ 6⎥ ⎥ 5⎥ ⎥ 4⎥ ⎥ 3⎦ 0
4 5 6 7 0 1 2 3
⎡
5 4 7 6 1 0 3 2
6 7 4 5 2 3 0 1
3
0 1 2 3 5 4 7 7 6
⎤
7 6⎥ ⎥ 5⎥ ⎥ 4⎥ ⎥ 3⎥ ⎥ 2⎥ ⎥ 1⎦ 0 1 2 0 4 3 6 5 8
⎢0 ⎢ ⎢1 ⎢ ⎢2 ⎢ ⎢4 ⎢ ⎢5 ⎢ ⎣6
33
2 4 3 5 4 3 0 1 1 0 7 2 8 9 5 10
5 4 6 7 2 0 1 3
6 7 5 8 9 1 0 2
⎤
7 6 ⎥ ⎥ 8 ⎥ ⎥ 5 ⎥ ⎥ 10 ⎥ ⎥ 3 ⎥ ⎥ 2 ⎦ 0
Fig. 2. A0 (7, 7) (above) and A2 (7, 7) and A3 (7, 7) (below, left and right).
⎡
0 ⎢1 ⎢ ⎢2 ⎢ ⎢3 ⎢ ⎢4 ⎢ ⎢5 ⎢ ⎣6 7
1 2 0 3 3 0 2 1 5 6 4 7 7 4 6 5
3 2 1 0 7 6 5 4
4 5 6 7 0 1 2 3
5 4 7 6 1 0 3 2
6 7 4 5 2 3 0 1
7 6 5 4 3 2 1 0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎡
2 ⎢0 ⎢ ⎢1 ⎢ ⎢3 ⎢ ⎢4 ⎢ ⎢5 ⎢ ⎣6 7
0 1 2 4 3 6 5 8
1 2 0 5 6 3 4 9
3 4 5 0 1 2 7 6
4 3 6 1 0 7 2 5
5 6 3 2 7 0 1 4
6 5 4 7 2 1 0 3
7 8 9 6 5 4 3 0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Fig. 3. Some examples of the Locator Theorem. In each array, the elements involved are boxed. Note that the relevant row and column indices are, for these examples, the corresponding entries in column 0 and row 0, respectively.
of Stromquist and Ullman gives rise to the full family of arrays As . Indeed, the (i , j )-entry of As is |(G 1 ⊕ G 2 ) → Nims |. The arrays As can be used to guide the actual play of a game (G 1 ⊕ G 2 ) → Nims ; we direct the reader to [1, §2] for more information. A fundamental property of As follows as an immediate consequence of the recursive construction (see [1] for details): Proposition 2.4. (See [1], Proposition 2.4.) For each s, the array As is symmetric, and each nonnegative integer appears exactly once in each row (and, by symmetry, each column). From this proposition, it easily follows that for each i ∈ N0 , the function mi : N0 → N0 defined by mi ( j ) := ai , j is a bijection. The main focus of this paper is the exploration of the following “Locator Property” in the arrays As for a variety of seeds s: Definition 2.5. For a fixed seed s we say that the Locator Property holds at (i , j ) if there is an integer k such that mi ( j ) = k and mi (k) = j, i.e., if, in the cycle decomposition of mi , the element j appears in a transposition. Equivalently, the Locator Property holds at (i , j ) if we have ai ,ai, j = j, i.e., in row i the entry equal to j is located in column ai , j . Several related illustrations of the Locator Theorem are highlighted in each array in Fig. 3. For seed s = 0 (in the lefthand array) we have, for example, a3,4 = m3 (4) = 7 and a3,7 = m3 (m3 (4)) = 4. Similarly, a4,7 = m4 (7) = 3 and a4,3 = m4 (m4 (7)) = 7. Turning now to seed s = 2 (in the right-hand array) we have a4,5 = m4 (5) = 7 and a4,7 = m4 (m4 (5)) = 5, as well as a7,4 = m7 (4) = 5 and a7,5 = m7 (m7 (4)) = 4. The reader can easily discern additional examples among the highlighted entries. We end this section with a recapitulation of some key definitions and propositions from our papers [1,2]. Definition 2.6. 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 2.7. By algorithm 2, which is well-defined only for principal subarrays As ( 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 = 0 and suppose that all entries less than k have been placed (if k = 0 then nothing other than the seed has been / Up(i , j ) and (i , j ) is not yet assigned an entry}; if m q then set the placed). Starting with row i = 0, let m = min{ j: k ∈
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(i , m) entry to k. Now increment i and, if i p, repeat the calculation of m. Carry out this process for all k, incrementing k each time, until all entries in As ( p , q) have been filled. Algorithm 2 succeeds in correctly filling out a finite portion of As because when computing mex( X ) for a set X , only those entries less than mex( X ) are actually relevant to the calculation. As proved in [2], the rows, columns, and diagonals in As do eventually settle into distinct periodic or arithmetically periodic patterns for s 2. These patterns are frequently complicated to compute, with the behavior in the first few rows and diagonals the easiest to find by hand. As an example, we give the following properties from [1], both of which will be needed below. For the reader who wishes to consult reference [1], which makes use of an operation ∗, we note that for small values of i , j we have mi ( j ) = i ∗ j, but for i , j > s we do have mi ( j ) = i ∗ j. Property 2.8. For all seeds s and all k,
a0,k = ak,0 =
⎧ ⎨s
k−1 ⎩ k
if k = 0 if 0 < k s if k > s.
Property 2.9. For all seeds s 1 and all k,
ak,k =
⎧ ⎨s 1 ⎩ 0
if k = 0 if k = 1 otherwise.
Property 2.9 may easily be verified using algorithm 2. 3. The Locator Theorem We first discuss seeds 0 and 1 since these behave very differently from other seeds. Proposition 3.1. 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, ai , j is found by bit-wise XOR [4]. Specifically, we can compute ai , j = i ⊕ j. Since bit-wise XOR is associative, we have mi (mi ( j )) = mi (i ⊕ j ) = i ⊕ (i ⊕ j ) = (i ⊕ i ) ⊕ j = 0 ⊕ j = j for all i , j, so the Locator Property holds for seed 0. Note now that A1 differs from A0 only in A1 (1, 1). Thus, in seed 1, as long as either i > 1 or j > 1 the same calculation as for seed 0 gives mi (mi ( j )) = j. It is simple to confirm that each of the remaining four entries in A1 (1, 1) also satisfies the Locator Property. 2 The proof of the remainder of our main result rests on the idea of a set of indices being Left- or Up-complete, which we define here. Even though they are similar, Left- and Up-completeness do not work quite the same. Definition 3.2. A set X of column-indices is i-Left-complete if, in row i, the entries equal to the elements of X are located in the columns indexed by mi ( X ) = {mi ( j ): j ∈ X }. A set Y of row-indices is j-Up-complete if, in the rows indexed by Y , the entries equal to j are located in the columns indexed by m Y ( j ) = {mi ( j ): i ∈ Y }. Before proceeding with an example, we remind the reader that rows and columns are indexed from 0. For example, in seed s = 3 (see Fig. 4), the set {0, 1, 2, 3} is 2-Left-complete because the entries in row 2 equal to 0, 1, 2, 3 are located in columns indexed by 2, 0, 1, 4 (respectively), and m2 ({0, 1, 2, 3}) = {1, 2, 0, 4}. Similarly, the set Y = {0, 1, 2, 3, 4} is 3-Up-complete in seed s = 3 because the entries equal to j = 3 in rows 0, 1, 2, 3, 4 are located in columns 0, 3, 4, 1, 2 (respectively) and m{0,1,2,3,4} (2) = {1, 2, 0, 4, 3}. Note that for Left-completeness, we are looking for the column locations of several values in a single row, whereas in Up-completeness, we are looking for the column locations of a single value in several rows. We note two important properties of Up-complete and Left-complete sets that readily follow from the definitions: Property 3.3. A union of i-Left-complete sets is i-Left-complete and a union of j-Up-complete sets is j-Up-complete. Property 3.4. If the Locator Property holds at (i , j ), then X = { j } is i-Left-complete and Y = {i } is j-Up-complete.
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Fig. 4. Illustrations in seed 3 of 2-Left-completeness (on the left) and 3-Up-completeness (on the right). On the left, the circled entries in row 2 are located in the columns whose indices are in the horizontal rectangle in row 2. On the right, the entries equal to 3 are in the columns whose indices are in the vertical rectangle in column 3.
Lemma 3.5. For all seeds s 2, 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-Upcomplete. 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). Proof. To demonstrate the first assertion we use Properties 2.8 and 2.9 repeatedly. That {0, 1, . . . , k} for k < s is not 0-Left-complete when s 2 follows by noting that if k < s then in row 0, the entry k − 1 occurs in column k ∈ / m0 ( X ) = {m0 (0), . . . , m0 (k)} = {s, 0, 1, . . . , k − 1}. However, if X = {0, 1, . . . , s} then m0 ( X ) = X , so we do have 0-Left-completeness. Let Y = {0, 1, . . . , k}. For k = 0 we have m Y (0) = {s}, but since s > 1 and a0,1 = 0, by Property 2.4, 0-Up-completeness fails. By Property 2.9, for k > 0 the entries equal to 0 in rows indexed by Y occur in the columns indexed by Y . Since we have Y = m Y (0) for k < s and Y = m Y (0) for k = s, 0-Up-completeness holds for k = 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 2.8 we have a0, j = j and ai ,0 = i. The second assertion readily follows. By Property 2.9, the third assertion also follows. 2 Lemma 3.6. 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 ) = {a0, 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 {a0, j , . . . , a(i −1), j }. Likewise, since {0, 1, . . . , ( j − 1)} is i-Left-complete, looking at Left(i , j ) = {ai ,0 , . . . , ai ,( j −1) }, we find the column numbers in which 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; those arising because j is already in that column (enumerated in Up(i , j ) in a row of smaller index), and those 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 is precisely the mex of the forbidden column numbers. Thus the entry j in row i goes into the column numbered m = mex(Left(i , j ) ∪ Up(i , j )) = ai , j . In other words, entry j in row i appears in column ai , j , so ai ,ai, j = j, i.e., mi (mi ( j )) = j. 2 Lemma 3.7. For seed s = 2, the set {0, 1, 2} is both i-Left-complete for i = 0, 1, 2 and j-Up-complete for j = 0, 1, 2. Lemma 3.8. For seed s = 3, the following hold: 1. 2. 3. 4. 5.
The set {0, 1, 2, 3} is i-Left-complete for i = 0, 1, 2 and 0-Up-complete. The set {0, 1, 2, 3, 4} is 3-Up-complete and 3-Left-complete. The set {1, 2, 3, 4, 5} is 4-Left-complete. The sets {1, 2} and {i − 1, i , i + 1} are i-Left-complete for all i 5. The set { j − 1, j , j + 1} is j-Up-complete for all j 4.
These lemmas can readily be proven by induction using Fig. 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 5 and 7, but the reader can easily surmise them.
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Fig. 5. The values
ρs for seeds s ∈ {4, 6, 8, 9, 10, . . .}.
Theorem 3.9 (The Locator Theorem). For seeds s = 2, 3, 5, 7, the Locator Property holds for all (i , j ) ∈ L s where L 2 = {(i , L 3 = {(i , L 5 = {(i , L 7 = {(i ,
j ): j ): j ): j ):
i > s or j > s}, (i > s or j > s) and j = 1, 2 and |i − j | > 1}, (i > s + 1 or j > s + 1) and j = 1, 2, 3, 4 and |i − j | > 2}, (i > s + 2 or j > s + 2) and j = 1, 2, 3, 4, 5 and |i − j | > 3}.
Proof. We use Lemmas 3.5, 3.7, and 3.8, as well as their analogues for seeds 5 and 7, as base cases. Suppose (i , j ) ∈ L s for s ∈ {2, 3, 5, 7}, and moreover that the Locator Property holds for all (i , j ) ∈ L s with i < i or j < j. By the base cases and Property 3.4 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 3.3, {0, 1, . . . , ( j − 1)} is i-Leftcomplete and {0, 1, . . . , (i − 1)} is j-Up-complete. Lemma 3.6 therefore proves that the Locator Property holds at (i , j ). 2 We now return to the game described in the introduction and depicted in Fig. 1. That game can be described as
(Nim j ⊕ Nimi ) → Nims ⊕ Nimi → Nims ⊕ Nim j , and the Grundy value of this game is exactly Nimmi (mi ( j )) ⊕ Nim j , where mi is defined in terms of the array As . Thus, the game is a second player win exactly when mi (mi ( j )) = j, i.e., when the Locator Property holds at (i , j ). The beginning of a winning strategy for the second player would be to mirror any move by the first player in an “i” pile in the other “i” pile, and similarly for the “ j” pile, as long as this returns the game to a configuration for which the Locator Property holds. 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 As , let
P s = (i , j ) ∈ {0, 1, 2, . . . , 2999}2 : j > s and s < ai , j < 3000 ,
Q s = (i , j ) ∈ P s : ai ,ai, j = j , and ρs = | Q s |/| P s |. We have
if s ∈ {0, 1, 2, 3, 5, 7} then ρs = 1; if s ∈ {4, 6} ∪ {8, 9, . . . , 200} then ρs ∈ (0.155, 0.279). Thus ρs gives a measure of the extent to which entries in As satisfy the Locator Property. Fig. 5 displays the values of ρs for seeds other than {0, 1, 2, 3, 5, 7}. Note that in the definition of P s the conditions j > s and ai , j > s reflect 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 3.7 and 3.8).
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Acknowledgements Thomas Zaslavsky offered helpful suggestions during the writing stage. In addition, comments from the anonymous referees improved the exposition significantly. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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