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Periodicity and Other Structure in a ColorfulFamily of Nim-like Arrays Submitted: May 21, 2009; Accepted: Jul 13, 2010; Published: Jul 20, 2010 Mathematics Subject Classification: 68R15,

Periodicity and Other Structure in a Colorful

Family of Nim-like Arrays

Submitted: May 21, 2009; Accepted: Jul 13, 2010; Published: Jul 20, 2010

Mathematics Subject Classification: 68R15, 91A46

Abstract

We study aspects of the combinatorial and graphical structure shared by acertain family of recursively generated arrays related to the operation of Nim-addition In particular, these arrays display periodic behavior along rows anddiagonals We explain how various features of computer-generated graphicsdepicting these arrays are reflections of the theorems we prove

Keywords: Nim, Sprague-Grundy, periodicity, sequential compound

1 Introduction

The game of Nim is a two-person combinatorial game consisting of one or more piles

of stones in which the players alternate turns removing any number of stones theywish from a single pile of stones; the winner is the player who takes the last stone.The direct sum G1 ⊕ G2 of two combinatorial games G1, G2 is the game in which a

∗ Partially supported by The Johns Hopkins University’s Acheson J Duncan Fund for the vancement of Research in Statistics

Ad-player, on their turn, has the option of making a move in exactly one of the games

G1 or G2 which are not yet exhausted (in Nim this simply means having severalindependent piles of stones) Again, the winner is the last player to make a move.The importance of Nim was established by the Sprague-Grundy Theorem [14, 24](also developed in [7, chapter 11]), which essentially asserts that Nim is universalamong finite, impartial two-player combinatorial games in which the winner is theplayer to move last Briefly, that is to say that every such game G is, vis-a-vis directsum, equivalent to a single-pile Nim game; we write |G| for the size of that singlepile, and call it the “Grundy-value” of G

In [26], Stromquist and Ullman define an operation on games called “sequentialcompound.” Essentially, the sequential compound G → H of games G and H is thegame in which play proceeds in G until it is exhausted, at which point play switches to

H In this paper we explore combinatorial games whose structure is (G1⊕ G2) → H,where G1, G2, and H are independent impartial combinatorial games Note that theGrundy value of (G1 ⊕ G2) → H is determined by the Grundy values of G1, G2,and H Previously, little was understood about this type of sequential compound inthe case that H is equivalent to a Nim-pile with more than one stone in it (if H isequivalent to a Nim-pile with one stone it in, this is called mis`ere play) Our resultshere cover sequential compounds of this type for piles of any size

The Sprague-Grundy Theorem implies that direct-sum of Nim-piles yields an eration, called Nim-addition, on N0 = {0, 1, 2, }, and it is well known that Nim-addition may be represented as a recursively generated array [4] The purpose of thispaper is to give a detailed combinatorial and graphical description of the members

op-of a family A∗ = {As}s∈N0 of related recursively generated arrays corresponding to acombination of direct sum and sequential compound The subscript s corresponds tothe Grundy-value of the game H; the array A0 is thus the Nim-addition table itself,and the array A1 arises from mis`ere play [4] The array A2 was first mentioned in[26], where Stromquist and Ullman commented that it “reveals many curiosities butfew simple patterns.” The results and observations in this paper were developed bythe authors to explain some of those many curiosities, not just for A2 but for all

As

Until recently, there appears to have been no other discussion in the literature of

A∗ or the “sequential compound” operation introduced in [26] which gave rise tothese arrays, other than a brief mention in a list of problems compiled by RichardGuy [15, Problem 41] Recently, however, Rice described each of the arrays As asendowing N0 with the algebraic structure of a quasigroup [22] We discuss resultsrelated to this algebraic approach in [1]; that article deals with the same family A∗,

but approaches it from a very different perspective than the one used here Evenmore recent is the article [25] which describes the monoid structure on the set ofall combinatorial games endowed by sequential compound (called there “sequentialjoin”).

In contrast to the situation for A∗, there has been a fair amount of discussion ing an array arising in the study of Wythoff’s game [27, 4, 5, 8, 17, 20, 23] Wythoff’sgame is played in a similar fashion to the game of Nim, but in Wythoff’s there areexactly two piles of stones and players may either take any number of stones from asingle pile of stones or take the same number of stones from both of the piles As inNim, the winner is the player who takes the last stone In the recent paper [23], Ricedefines a family of arrays W∗ = {Ws}s∈N0 generalizing Wythoff’s game in essentiallythe same way as A∗ generalizes Nim Some of the ideas used in that context transferfairly readily to A∗; this is the case, for instance, with our Row Periodicity Theorem4.1 below

regard-The study of tables of Grundy values for various combinatorial games has led someresearchers to speak of “chaotic” behavior [4, 8, 11, 12, 28] As Zeilberger says,

“it seems that we have ‘chaotic’ behavior, but in a vague, yet-to-be-made-precise,sense.” [28] Part of this story is simply the hard-to-fathom distribution of values

in these tables Another aspect of it, though, is the availability of a variety ofperiodicity results, as in [2, 4, 5, 6, 8, 13, 16, 17, 23, 22, 28] The recent work

of Friedman and Landsberg on interpreting combinatorial games in the context ofdynamical system theory contributes yet another perspective [11, 12] Our paper,through combinatorial results about periodicity and other structural features, aims

to explain some of the complexity of the arrays As One result of all of this is theheightening of the expectation that there is indeed a precise sense in which thesearrays display behavior which is “chaotic.”

We open our discussion by providing, in Section 2, two different algorithms for structing the arrays As In addition, we include two lemmas describing the locations

con-of the entry 0 in the arrays and also the entries which occur in row 0

In Section 3 we begin our analysis by coloring the arrays A0 and A2 using a greenand purple scheme (see Figures 3 and 4) Although we focus on A2 rather than anyother As for s > 2, we have checked the colorings for s = 3, 4, , 100 and they areall quite similar Moreover, the various theorems we prove in this article, which holdfor every seed s > 2, show that this is to be expected It is interesting that theseresults seem to provide evidence for the conjecture in [11] that “generic, complexgames will be structurally stable.”

The array A0 has a very regular structure Although A2 may, at first glance, seemcompletely irregular, on more careful study one can see that it also has some definitestructure In Figure 5 we note three distinct [classes of] regions in the coloring of

A2:

1 The region of elements in green along the main diagonal (“spindle”)

2 Other regions of green, all starting from the top left corner (“tendrils”)

3 All other regions, mostly in purple (“background”)

This coloring, and similar ones for the other arrays As, give strong empirical dence that the arrays are highly structured We offer in this paper a more formalframework for making sense of these empirical observations: We provide an intrinsiccharacterization of these regions and then identify, in Proposition 3.1, Proposition3.3, and Theorem 3.4, some of the specific properties they enjoy We end Section

evi-3 with a coloring of the Wythoff array W0 which highlights some major differencesbetween that game and the arrays As

The overall complexity revealed by the coloring scheme bolsters the sense that ing As through the approach of combinatorial game theory would be quite difficult

analyz-A much more productive alternative is to see analyz-As in the context of combinatorics onwords (see [18, 19], for example) An n-dimensional word is a function from Zn or Nn

to some alphabet, and thus As and its subarrays may be viewed as two-dimensionalwords over the alphabet of nonnegative integers, and its rows and columns as one-dimensional words A fundamental notion in the study of words is periodicity (see[19, Chapter 8]), and this plays an important role in the study of As

Most work on periodicity, and indeed in combinatorics on words in general, has dealtwith one-dimensional words, but some attention has been paid to higher dimensions

In [3], Amir and Benson introduce notions of periodicity for two-dimensional words.More recently, [10] and [21] generalize some well-known one-dimensional periodicitytheorems to two dimensions As described in Section 4, the array As displays a fas-cinating interplay between periodicity in dimensions one and two On the one hand,Theorem 4.1 (“Row Periodicity”) asserts that there is a periodicity inherent in therows, and hence columns On the other hand, Theorem 4.4 (“Diagonal Periodicity”)describes periodicity in the placement, relative to the diagonal, of entries of a specificvalue

We conclude in Section 5 with a compelling computational observation, dently observed in a footnote in [11], that the array possesses a type of 2-fold scal-ing We formulate this in Conjecture 5.1 Additionally, we formulate some ques-

indepen-tions concerning the structure of the tendrils and background These require furtherstudy.

2 Mex and the Arrays As

In the following definitions, and in other material through Figures 1 and 2 andProposition 2.5, we closely follow [1] We begin by constructing a family of infinitearrays using the mex operation:

Definition 2.1 For a set X of non-negative integers we define mex X to be thesmallest non-negative integer not contained in X Here, mex stands for minimalexcluded value

Definition 2.2 For any 2-dimensional array A indexed by N0, let ai,j denote theentry in row i, column j, where i, j > 0 The principal (i, j) subarray A(i, j) isthe subarray of A consisting of entries ap,q with indices (p, q) ∈ {0, , i}×{0, , j}.For j > 0 define Left(i, j)= {ai,q : q < j} to be the set of all entries in row i tothe left of the entry ai,j, and for i > 0 define Up(i, j)= {ap,j : p < i} to be theset of entries in column j above ai,j (Note that Left(i, 0)=Up(0, j)=∅.) Also, defineDiag(i, j) to be {ai0 ,j 0 : i0 < i and i0− j0 = i − j}

Definition 2.3 The infinite array As, for s ∈ N0, is constructed recursively: Theseed a0,0 is set to s and for (i, j) 6= (0, 0),

ai,j := mex Left(i, j) ∪ Up(i, j)

See, for example, Figures 1 and 2 We note that in all figures the index i increasesgoing down the page and j increases going to the right

The reader can easily verify that a change of seed from 0 to 1 has a minimal effect;other than the top left 2 × 2 block, the pattern of the array A1 is exactly the same

Having the arrays As in hand has a direct usefulness when playing a game (G1 ⊕

G2) → ∗s A Grundy-value of 0 indicates that the “previous” player to move (i.e., theplayer who is not making the next move) has a winning strategy, and any nonzeroGrundy-value indicates that the next player to move has a winning strategy If

|G1| = i and |G2| = j, then for each a ∈ Up(i, j) there is a move in G1 (depending onthe specifics of G1) that results in a new game G01 such that |(G01⊕ G2) → ∗s| = a.Similarly, for a ∈ Left(i, j) there is a move in G2 that results in a new game G02 suchthat |(G1⊕ G0

2) → ∗s| = a

We present some of the practical implications: If s = 0 and |G1| < |G2| then amove in G = (G1 ⊕ G2) → ∗s which leaves G1 alone and changes G2 to a gamewith Grundy-value |G1| is a winning move If s > 0 and 1 < |G1| < |G2| then thesame is true, but when |G1| = 1 the winning move is to change G2 to a game withGrundy-value 0, and when |G1| = 0 the winning move is to change G2 to a gamewith Grundy-value 1

It may appear that only the location of the 0 values in As is of concern for playing, but this is not the case To see that the full information of the array As

game-is useful, consider games of the form (G1 ⊕ G2) → ∗s⊕ G3 In this case, awinning move in (G1 ⊕ G2) → ∗s could be a losing move overall (for instance, if

G3 is a single Nim-pile) On the other hand, a move in G1 to a game G01 such that

|(G0

1⊕ G2) → ∗s| = |G3|, for example, would be a winning move, and thus knowledge

of the locations of entries in As equal to |G3| is quite useful

in-While this holds for A0 and A2 equally, it is evident from Figure 2 that A2 is not

at all predictably regular, in direct contrast to A0 Although the entries in A0 can

be calculated directly (i.e., non-recursively) using bit-wise XOR [4], where the kthbinary digit of ai,j in A0 is equal to 1 if exactly one of i or j has a 1 in the kthbinary place, we have not found any non-recursive way to calculate entries of As forany s > 2 (and suspect that such an algorithm does not exist) There are, however,two different recursive algorithms which may be used to compute As As each hasits own advantages, we record them here:

Definition 2.5 When we refer to algorithm 1, we mean the algorithm describedabove in Definition 2.3, using the mex operation to fill in increasingly large subarrayscontaining the seed

Definition 2.6 In algorithm 2, which is well-defined only for principal subarrays

As(p, q), first all 0’s are filled in, then all 1’s, then all 2’s, etc Begin with the seed

Figure 3: The coloring of A0(511, 511) by size of entry; green represents smallervalues and purple represents larger.

s in the upper left hand corner and, starting with k = 0, suppose that all entries lessthan k have been placed (if k = 0 then nothing other than the seed has been placed).Starting with row i = 0, let m = min{j : k 6∈ Up(i, j), k 6∈ Left(i, j), and the (i, j)entry is not yet assigned an value; if m 6 q then set the (i, m) entry to k Nowincrement i and, if i 6 p, repeat the min calculation Otherwise, increment k andrepeat the process from the beginning (i.e., starting at i = 0), until all entries in

As(p, q) have been filled

Algorithm 2 succeeds in correctly filling out a finite portion of As because whencomputing mexX for a set X, only those entries less than mexX are actually relevant

to the calculation An analogue of algorithm 2 for Wythoff’s game is described in[5], where it is termed “Algorithm WSG.”

We end this section with two lemmas needed in later sections – the first describingthe patterns in row zero, and the second describing the placement of entries equal

to zero These were independently proven in [22]

3 Visualizing As

The regularity in A0 becomes striking when we assign colors to the entries usinggreen for the smallest values and purple for the largest values (and interpolatinglinearly between) For the principal subarray A0(511, 511) we obtain the image inFigure 3

The array A2, on the other hand, has a much more complicated structure We color

A2(1200, 1200) using the same green and purple scheme as for A0 (i.e., green →smallest, purple → largest); see Figure 4 It seems that pictures for As with s > 3are very similar to that of A2; we have checked this for s = 3, 4, 100

In A2, there seem to be three distinct colored regions: The elements in green alongthe main diagonal form a region which we will refer to as the “spindle.” There areother regions of green, all extending from the top left corner down and to the right;these will be referred to as “tendrils.” All other regions (in purple, mostly), will bereferred to as the “background.” In fact, we can identify these regions; for any s,partition the coordinate pairs for Asinto three sets and assign colors as follows:

S := {(i, j) : ai,j 6 min(i, j)} ← Red

T := {(i, j) : min(i, j) < ai,j 6 max(i, j)} ← Gold

B := {(i, j) : max(i, j) < ai,j} ← Grey

We will speak of this as a “partition of As,” and of S, T , and B as if they are blocks

of this partition of As Coloring A2(1200, 1200) using the above scheme yields Figure5

Compare this to the original green and purple picture in Figure 4; the red entriesdefine the spindle (S), the gold entries define the tendrils (T ), and the grey entriesdefine the background (B)

Taking another tack, we again color A2(1200, 1200) using the red, gold, grey scheme

as above, but this time adjust the shading of each color by the quantity of 1’s in the

Figure 4: The coloring of A2(1200, 1200) by size of entry; green represents smallervalues and purple represents larger.

Figure 5: The color-coded partition of A2(1200, 1200) into S (red), T (gold), and B(grey)