On Remoteness Functions of k-NIM with k + 1 Piles in Normal and in Misère Versions

Abstract

Given integer n and k such that $0<k\le n$ and n piles of stones, two players alternate turns. On each move, a player is allowed to choose any k piles and remove exactly one stone from each. The player who has to move but cannot is the loser in the normal version of the game and (s)he is the winner in the misère version. Cases $k=1$ and $k=n$ are trivial. For $k=2$, the game was solved for $n\le 6$. For $n\le 4$, the Sprague–Grundy function was efficiently computed (for both versions). For $n=5,6$, a polynomial algorithm computing P-positions was obtained for the normal version. Then, for the case $k=n-1$, a very simple explicit rule that determines the Smith remoteness function was found for the normal version of the game: the player who has to move keeps a pile with the minimum even number of stones; if all piles have an odd number of stones, then (s)he keeps a maximum one, while the $n-1$ remaining piles are reduced by one stone each in accordance with the rules of the game. Computations show that the same rule works efficiently for the misère version too. The exceptions are sparse. We list some. Denote a position by $x=(x_1,\dots ,x_n)$. Due to symmetry, we can assume wlog that $x_1\le \dots \le x_n$. Our computations partition all exceptions into the following three families: $x_1$ is even, $x_1=1$, and odd $x_1\ge 3$. In all three cases, we suggest formulas covering all found exceptions, but it is not proven that there are no others.

1. Introduction

We assume that the reader is familiar with basic concepts of impartial game theory (see e.g., [1,2,3,4] for an introduction) and also with the recent paper [5], where the normal version of game NIM$(n,k)$, the exact slow NIM, was analyzed for the case $n=k+1$. Here, we consider the misère version for this case.

1.1. Exact Slow NIM

Game Exact Slow NIM was introduced in [6] as follows: Given two integers n and k such that $0<k\le n$, and n piles containing $x_1,\dots ,x_n$ stones each. On each move, a player is allowed to reduce any k piles by exactly one stone each. Two players alternate turns. A player who has to move but cannot is the loser in the normal version of the game and (s)he is the winner in the misère version. In [6], this game was denoted ${\mathrm{NIM}}_{=}^1(n,k)$. Here, we will simplify this notation to NIM$(n,k)$.

Game NIM$(n,k)$ is trivial if $k=1$ or $k=n$. In the first case, it ends after $x_1+\dots +x_n$ moves and in the second one—after $min(x_1,\dots ,x_n)$ moves. In both cases, nothing depends on the players’ skills. All other cases are more complicated.

The game was solved for $k=2$ and $n\le 6$. In [7], an explicit formula for the Sprague–Grundy (SG) function was found for $n\le 4$, for both the normal and misère versions. This formula allows us to compute the SG function in linear time. Then, in [8], the P-positions of the normal version were found for $n\le 6$. For the subgame where $x_1+\dots +x_n$ is even, a simple formula for the P-positions was obtained, allowing verifification in linear time if x is a P-position and, if not, finding a move from it to a P-position. The subgame with odd $x_1+\dots +x_n$ is more difficult. Still, a (more sophisticated) formula for the P-positions was found, providing a linear time recognition algorithm.

Further generalizations of exact slow NIM were considered in [9].

1.2. Case $n=k+1$, the Normal Version

In [5], the normal version was solved in case $n=k+1$ by the following simple rule:

• (o) if all piles are odd, keep a largest one and reduce all others;
• (e) if there exist even piles, keep the smallest one of them and reduce all others.

This rule is well-defined and it uniquely determines a move in every position x. (Obviously, permuting the piles with the same number of stones, we keep the game unchanged.) The rule and the corresponding moves are called the M-rule and M-moves; the sequence of successive M-moves is called the M-sequence.

Obviously, $n>1$ is required. If $n=1$, then $x_1$ will reach an even value in at most one M-move, after which it will stop. Since this case is trivial, we can assume that $n>1$ without any loss of generality (wlog).

It is also easily seen that no M-move can result in a position whose entries are all odd. Hence, for an M-sequence, part (o) of the M-rule can be applied at most once, at the beginning; after this, only part (e) works.

Given a position $x=(x_1,\dots ,x_n)$, assume that both players follow the M-rule and denote by $\mathcal{M}\left(x\right)$ the number of moves from x to a terminal position. In [5], it was proven that $\mathcal{M}=\mathcal{R}$, where $\mathcal{R}$ is the classical remoteness function introduced by Smith [10]. Thus, the M-rule solves the game and, furthermore, allows a player to win as quickly as possible in an N-position and to resist as long as possible in a P-position.

A polynomial algorithm computing $\mathcal{M}=\mathcal{R}$ (and in particular, the P-positions) is given, even if n is a part of the input and integers are presented in binary form.

Let us also note that an explicit formula for the P-positions is known only for $n\le 4$, and for $n=3$, it is already quite complicated [7] and the Appendix in [5].

1.3. Related Versions of NIM

By definition, the present game NIM$(n,k)$ is the exact slow version of the famous Moore’s ${\mathrm{NIM}}_k$ [11]. In the latter game, a player, by one move, reduces arbitrarily (not necessarily by one stone) at most k piles from n.

The case $k=1$ corresponds to the classical NIM whose P-position was found by Bouton [12] for both the normal and misère versions.

Remark 1.

Actually, the Sprague–Grundy (SG) values of NIM were also computed in Bouton’s paper, although were not defined explicitly in general. This was completed later by Sprague [13] and Grundy [14] for arbitrary disjunctive compounds of impartial games; see also [3,10].

In fact, the concept of a P-position was also introduced by Bouton in [12], but only for the (acyclic) digraph of NIM, not for all impartial games. In its turn, this is a special case of the concept of a kernel, which was introduced for arbitrary digraphs by von Neumann and Morgenstern [15].

Also the misère version was introduced by Bouton in [12], but only for NIM, not for all impartial games. The latter was completed by Grundy and Smith [16]; see also [3,10].

Moore [11] obtained an elegant explicit formula for the P-positions of ${\mathrm{NIM}}_k$ generalizing the Bouton’s case $k=1$. Even more generally, the positions of the SG-values 0 and 1 were efficiently characterized by Jenkins and Mayberry [17]; see also Section 4 in [18].

Also in [17]; the SG function of ${\mathrm{NIM}}_k$ was computed explicitly for the case $n=k+1$ (in addition to the case $k=1$). In general, no explicit formula, nor even a polynomial algorithm, computing the SG-values (larger that 1) is known. The smallest open case: 2-values for $n=4$ and $k=2$.

The remoteness function of k-NIM was recently studied in [19].

Let us also mention the exact (but not slow) game ${\mathrm{NIM}}^{=}(n,k)$ [18] in which exactly k from n piles are reduced (by an arbitrary number of stones) in a move. The SG-function was efficiently computed in [18] for $n\le 2k$. Otherwise, even a polynomial algorithm looking for the P-positions is not known (unless $k=1$, of course). The smallest open case is $n=5$ and $k=2$.

2. Case $\mathit{n}=\mathit{k}+\mathbf{1}$, Misère Version

Computations show that the same M-rule works pretty efficiently also for the misère version of the considered game NIM$(n,k)$ with $n=k+1$, yet, not always. A position $x=(x_1,\dots ,x_n)$ is called an exception if the M-move is not optimal, in other words, if $\mathcal{R}\left(x\right)-\mathcal{R}\left(x^{\prime }\right)\ne 1$ for the M-move $x\to x^{\prime }$.

In fact, $\mathcal{R}\left(x\right)$ is odd and $\mathcal{R}\left(x\right)-\mathcal{R}\left(x^{\prime }\right)$ takes values 0 or 2 for all known exceptions. ($\mathcal{R}\left(x\right)$ and $\mathcal{R}\left(x^{\prime }\right)$ can be equal, but both cannot be even.) The exceptions are sparse and satisfy a regular pattern based on two parameters: n and $min(x_1,\dots ,x_n)$. However, the complete description of this pattern is open.

2.1. Monotonicity for the Entries of Positions

Recall that $x_1\le \dots \le x_n$ is assumed for any position $x=(x_1,\dots ,x_n)$ (We order the entries $x_i$ just for convenience; their permutations do not change the game.).

However, even if this monotonicity holds for x, it may fail for $x^{\prime }$, after a move $x\to x^{\prime }$. In this case, we have to restore it by permuting entries of $x^{\prime }$.

Alternatively, we can make the M-rule slightly stricter, as follows. Given a position $x=(x_1,\dots ,x_n)$ for which the M-rule is “ambiguous”; that is, x contains

• (o) several smallest even entries, or
• (e) several largest odd entries, provided all $x_i$ are odd.

In both cases, among these equal entries, keep one with the largest index reducing all others by 1. We will call such an M-move (as well as the corresponding M-sequence and M-rule) strict. It is easily seen that a strict M-move $x\to x^{\prime }$ respects the non-decreasing monotonicity of the entries; that is, $(x_1^{\prime }\le \dots \le x_n^{\prime })$ whenever $(x_1\le \dots \le x_n)$. In contrast, every non-strict M-move breaks this monotonicity.

2.2. Monotonicity of Exceptions

A position is called an exception if the M-move is not optimal in it. An optimal move in such a position is called exceptional.

Proposition 1.

Given integer $m\ge n$, a position $x^{\prime }=(x_1,\dots ,x_n,\dots x_m)$ is an exception whenever $x=(x_1,\dots ,x_n)$ exists; of course, not vice versa. Moreover, the exceptional moves coincide in x and $x^{\prime }$. (More precisely, the entry $x_i$ that is kept unchanged by an optimal move is the same for x and $x^{\prime }$; furthermore, $1\le i\le n$)

Listing the exceptional position below, by default, we do not include $x^{\prime }$ if x is already listed; in other words, we include only the minimal exceptions.

2.3. General Properties of Exceptions

All found minimal monotone exceptions share the following properties:

• [$x_n$-monotone] $x_n>x_{n-1}$; if position $x^i=(x_1,\dots ,x_{n-1},x_{n-1}+i)$ is an exception for some $i>0$, then $x^i$ is an exception for each $i>0$.
• $x_n-x_{n-1}=1$ in every minimal exception.
• [$x_{n-1}$-determining] $\mathcal{R}\left(x\right)=f\left(x_{n-1}\right)+1$, where $f(\ell )=2\lceil \ell /2\rceil$; that is, $f(\ell )=\ell$ if ℓ is even and $f(\ell )=\ell +1$ if ℓ is odd, for all integer $\ell \ge 0$.
  Thus, $\mathcal{R}\left(x\right)$ (and $\mathcal{R}\left(x^{\prime }\right)$) are odd (and, hence, the first player wins) in every exception. However, the M-move is losing. (It could win but would require a larger number of moves. Yet, such a case is not realized in any found exception.)
• In every exception x, the optimal move keeps the entry $x_n$ if $x_{n-1}$ is even and keeps $x_{n-1}$ if it is odd. In contrast, the strict M-move, vice versa, keeps $x_{n-1}$ if it is even and keeps $x_n$ if $x_{n-1}$ is odd.
• [$0\le \mathcal{R}\left(x\right)-\mathcal{R}\left(x^{\prime }\right)\le 2$] By definition of the remoteness function, we have $\mathcal{R}\left(x\right)-\mathcal{R}\left(x^{\prime }\right)=1$ for each optimal move $x\to x^{\prime }$ in every impartial game, in particular, for every M-move in the normal version of NIM$(n,n-1)$. In contrast, for its misère version, in every found minimal exception, $\mathcal{R}\left(x\right)-\mathcal{R}\left(x^{\prime }\right)$ takes only values 0, when $x_{n-1}$ is even, or 2, when $x_{n-1}$ is odd.

2.4. Even $x_1$

Given $x_1=2i$, a position $x=(x_1,\dots ,x_n,\dots x_m)$ is an exception if and only if

$$2i=x_1=\dots =x_{i+2}<x_{i+3}=x_n\le \dots \le x_m\mathrm{where}1\le i\le n-3\mathrm{and}n\ge 4.$$

Since $x_{n-1}$ is always even, the optimal move keeps $x_n$, while the M-move keeps $x_j$ for any fixed $j<n$. Since $\mathcal{R}\left(x\right)=2i+1$ is odd, the first player always wins. Note that both properties agree with Section 2.3.

Examples for $x_1=2,4,6,8$ are given below

$$\begin{array}{lll}2=x_1=x_2=x_3<x_4=x_n\le \cdots \le x_m,& \mathcal{R}\left(x\right)=3,& n=4;\\ 4=x_1=\dots =x_4<x_5=x_n\le \cdots \le x_m,& \mathcal{R}\left(x\right)=5,& n=5;\\ 6=x_1=\dots =x_5<x_6=x_n\le \cdots \le x_m,& \mathcal{R}\left(x\right)=7,& n=6;\\ 8=x_1=\dots =x_6<x_7=x_n\le \cdots \le x_m,& \mathcal{R}\left(x\right)=9,& n=7,\mathrm{etc}.\end{array}$$

2.5. $x_1=1$

A position $x=(x_1,\dots ,x_n,\dots ,x_m)$ with $x_1=1$ is an exception if and only if

$$1=x_1\le x_2<x_3\le \dots \le x_m,\mathrm{here}3=n\le m.$$

Furthermore, if $x_2$ is even, then $\mathcal{R}\left(x\right)=x_2+1$, and the only optimal move keeps $x_3$, while the M-move keeps $x_2$; if $x_2$ is odd, then $\mathcal{R}\left(x\right)=x_2+2$, and, in contrast, the only optimal move keeps $x_2$, while the M-move keeps $x_i$, for some $i>2$. In both cases, $\mathcal{R}\left(x\right)$ is odd and, hence, the first player always wins. It is easily seen that all these properties agree with Section 2.3.

Wlog, we could restrict ourselves by $n=3$. All exceptions with larger n are implied by monotonicity.

Thus, it remains to consider odd values of $x_1\ge 3$. Our computer analysis includes only $x_1=3,5,7,9,11,13,15,17$. In each case, we observe a pattern; however, its extension to arbitrary odd $x_1$ remains an open problem.

2.6. Odd $x_1\ge 5$ with $n=4$

For any odd $x_1\ge 5$, fix an integer $i\ge 0$ to obtain the following two exceptions $x=(x_1,x_2,x_3,x_4)$:

$$x_1,x_2=x_1+2i,x_3=2(x_1+i-2),x_4=2x_1+2i-3=x_3+1;$$

$$x_1,x_2=x_1+2i,x_3=2x_1+2i-3,x_4=2(x_1+i-1)=x_3+1.$$

In the first case, $\mathcal{R}\left(x\right)=x_4$, and the unique optimal move keeps $x_4$, while the unique M-move keeps $x_3$; in contrast, in the second case, $\mathcal{R}\left(x\right)=x_4+1$, and the unique optimal move keeps $x_3$, while the unique M-move keeps $x_4$.

Furthermore, the remoteness function is given by formula $\mathcal{R}\left(x\right)=f(x_3+1)$, where $f\left(m\right)=2\lfloor m/2\rfloor +1$; that is, $f\left(m\right)=m$ if m is odd and $f\left(m\right)=m+1$ if m is even, for all $m\ge 0$.

It is easily seen that $\mathcal{R}\left(x\right)$ is odd for each i; hence, the first player always wins.

Examples for $x_1=5,7,9,11$ are given below. Notation $y+$ means “any number that is greater than or equal to y”.

$x_1=5$	$\mathcal{R}$	$x_1=7$	$\mathcal{R}$	$x_1=9$	$\mathcal{R}$	$x_1=11$	$\mathcal{R}$
(5, 5, 6, 7+)	7	(7, 7, 10, 11+)	11	(9, 9, 14, 15+)	15	(11, 11, 18, 19+)	19
(5, 5, 7, 8+)	9	(7, 7, 11, 12+)	13	(9, 9, 15, 16+)	17	(11, 11, 19, 20+)	21
(5, 7, 8, 9+)	9	(7, 9, 12, 13+)	13	(9, 11, 16, 17+)	17	(11, 13, 20, 21+)	21
(5, 7, 9, 10+)	11	(7, 9, 13, 14+)	15	(9, 11, 17, 18+)	19	(11, 13, 21, 22+)	23
(5, 9, 10, 11+)	11	(7, 11, 14, 15+)	15	(9, 13, 18, 19+)	19	(11, 15, 22, 23+)	23
(5, 9, 11, 12+)	13	(7, 11, 15, 16+)	17	(9, 13, 19, 20+)	21	(11, 15, 23, 24+)	25

By monotonicity, any such exception $x=(x_1,x_2,x_3,x_4)$ can be extended to the exceptions $x^{\prime }=(x_1^{\prime },\dots ,x_m^{\prime })$, with $m\ge 5$ and $x_i=x_i^{\prime }$ for $i\le 4$, while $x_5^{\prime },\dots x_m^{\prime }$ can be chosen arbitrary such that $x_4\le x_5^{\prime }\le \dots \le x_m^{\prime }$. Note also that case $x_1=3$ is considered in Section 2.9.

2.7. Odd $x_1\ge 7$ with $n=5$

The following families of exceptions were found:

$x_1=7$	$x_1=9$	$x_1=11$	$x_1=13$
(7, 7, 8, 8, 9)	(9, 9, 12, 12, 13)	(11, 11, 16, 16, 17)	(13, 13, 20, 20, 21)
(7, 7, 9, 9, 10)	(9, 9, 13, 13, 14)	(11, 11, 17, 17, 18)	(13, 13, 21, 21, 22)
(7, 9, 10, 10, 11)	(9, 11, 14, 14, 15)	(11, 13, 18, 18, 19)	(13, 15, 22, 22, 23)
(7, 9, 11, 11, 12)	(9, 11, 15, 15, 16)	(11, 11, 19, 19, 20)	(13, 15, 23, 23, 24)
(7, 11, 12, 12, 13)	(9, 13, 16, 16, 17)	(11, 15, 20, 20, 21)	(13, 17, 24, 24, 25)

$x_1=9$	$x_1=13$	$x_1=17$	$x_1=21$
(9, 9, 9, 10, 11+)	(13, 13, 13, 16, 17+)	(17, 17, 17, 22, 23+)	(21, 21, 21, 28, 29+)
(9, 9, 9, 11, 12+)	(13, 13, 13, 17, 18+)	(17, 17, 17, 23, 24+)	(21, 21, 21, 29, 30+)
(9, 11, 11, 12, 13+)	(13, 15, 15, 18, 19+)	(17, 19, 19, 24, 25+)	(21, 23, 23, 30, 31+)
(9, 11, 11, 13, 14+)	(13, 15, 15, 19, 20+)	(17, 19, 19, 25, 26+)	(21, 23, 23, 31, 32+)
(9, 13, 13, 14, 15+)	(13, 17, 17, 20, 21+)	(17, 21, 21, 26, 27+)	(21, 25, 25, 32, 33+)

$x_1=11$	$x_1=13$	$x_1=15$	$x_1=17$
(11, 11, 13, 14, 15+)	(13, 13, 17, 18, 19+)	(15, 15, 21, 22, 23+)	(17, 17, 25, 26, 27+)
(11, 11, 13, 15, 16+)	(13, 13, 17, 19, 20+)	(15, 15, 21, 23, 24+)	(17, 17, 25, 27, 28+)
(11, 13, 15, 16, 17+)	(13, 15, 19, 20, 21+)	(15, 17, 23, 24, 25+)	(17, 19, 27, 28, 29+)
(11, 13, 15, 17, 18+)	(13, 15, 19, 21, 22+)	(15, 17, 23, 25, 26+)	(17, 19, 27, 29, 30+)
(11, 15, 17, 18, 19+)	(13, 17, 21, 22, 23+)	(15, 19, 25, 26, 27+)	(17, 21, 29, 30, 31+)

$x_1=15$	$x_1=17$	$x_1=19$	$x_1=21$
(15, 15, 17, 20, 21+)	(17, 17, 21, 24, 25+)	(19, 19, 25, 28, 29+)	(21, 21, 29, 32, 33+)
(15, 15, 17, 21, 22+)	(17, 17, 21, 25, 26+)	(19, 19, 25, 29, 30+)	(21, 21, 29, 33, 34+)
(15, 17, 19, 22, 23+)	(17, 19, 23, 26, 27+)	(19, 21, 27, 30, 31+)	(21, 23, 31, 34, 35+)
(15, 17, 19, 23, 24+)	(17, 19, 23, 27, 28+)	(19, 21, 27, 31, 32+)	(21, 23, 31, 35, 36+)
(15, 19, 21, 24, 25+)	(17, 21, 25, 28, 29+)	(19, 23, 29, 32, 33+)	(21, 25, 33, 36, 37+)

2.8. Odd $x_1\ge 9$ with $n=6$

The following families of exceptions were found:

$x_1=9$	$x_1=11$	$x_1=13$	$x_1=15$
(9, 9, 10, 10, 10, 11+)	(11, 11, 14, 14, 14, 15+)	(13, 13, 18, 18, 18, 19+)	(15, 15, 22, 22, 22, 23+)
(9, 9, 11, 11, 11, 12+)	(11, 11, 15, 15, 15, 16+)	(13, 13, 19, 19, 19, 20+)	(15, 15, 23, 23, 23, 24+)
(9, 11, 12, 12, 12, 13+)	(11, 13, 16, 16, 16, 17+)	(13, 15, 20, 20, 20, 21+)	(15, 17, 24, 24, 24, 25+)
(9, 11, 13, 13, 13, 14+)	(11, 13, 17, 17, 17, 18+)	(13, 15, 21, 21, 21, 22+)	(15, 17, 25, 25, 25, 26+)
(9, 13, 14, 14, 14, 15+)	(11, 15, 18, 18, 18, 19+)	(13, 17, 22, 22, 22, 23+)	(15, 19, 26, 26, 26, 27+)

$x_1=11$	$x_1=15$	$x_1=19$	$x_1=23$
(11, 11, 11, 12, 12, 13+)	(15, 15, 15, 18, 18, 19+)	(19, 19, 19, 24, 24, 25+)	(23, 23, 23, 30, 30, 31+)
(11, 11, 11, 13, 13, 14+)	(15, 15, 15, 19, 19, 20+)	(19, 19, 19, 25, 25, 26+)	(23, 23, 23, 31, 31, 32+)
(11, 13, 13, 14, 14, 15+)	(15, 17, 17, 20, 20, 21+)	(19, 21, 21, 26, 27, 27+)	(23, 25, 25, 32, 32, 33+)
(11, 13, 13, 15, 16, 16+)	(15, 17, 17, 21, 21, 22+)	(19, 21, 21, 27, 27, 28+)	(23, 25, 25, 33, 33, 34+)
(11, 15, 15, 16, 16, 17+)	(15, 19, 19, 22, 22, 22+)	(19, 23, 23, 28, 29, 29+)	(23, 27, 27, 34, 34, 35+)

$x_1=15$	$x_1=17$	$x_1=19$
(15, 15, 17, 17, 18, 19+)	(17, 17, 21, 21, 22, 23+)	(19, 19, 25, 25, 26, 27+)
(15, 15, 17, 17, 19, 20+)	(17, 17, 21, 21, 23, 24+)	(19, 19, 25, 25, 27, 28+)
(15, 17, 19, 19, 20, 21+)	(17, 19, 23, 23, 24, 25+)	(19, 21, 27, 27, 28, 29+)
(15, 17, 19, 19, 21, 22+)	(17, 19, 23, 23, 25, 26+)	(19, 21, 27, 27, 29, 30+)
(15, 19, 21, 21, 22, 23+)	(17, 21, 25, 25, 26, 27+)	(19, 23, 29, 29, 30, 31+)

$x_1=13$	$x_1=15$	$x_1=17$	$x_1=19$
(13, 13, 15, 16, 16, 17+)	(15, 15, 19, 20, 20, 21+)	(17, 17, 23, 24, 24, 25+)	(19, 19, 27, 28, 28, 29+)
(13, 13, 15, 17, 17, 18+)	(15, 15, 19, 21, 21, 22+)	(17, 17, 23, 25, 25, 26+)	(19, 19, 27, 29, 29, 30+)
(13, 15, 17, 18, 18, 19+)	(15, 17, 21, 22, 22, 23+)	(17, 19, 25, 26, 26, 27+)	(19, 21, 29, 30, 30, 31+)
(13, 15, 17, 19, 19, 20+)	(15, 17, 21, 23, 23, 24+)	(17, 19, 25, 27, 28, 28+)	(19, 21, 29, 31, 31, 32+)
(13, 17, 18, 20, 20, 21+)	(15, 19, 21, 24, 24, 25+)	(17, 21, 27, 28, 28, 29+)	(19, 23, 31, 32, 32, 33+)

$x_1=13$	$x_1=19$	$x_1=25$
(13, 13, 13, 13, 14, 15+)	(19, 19, 19, 19, 22, 23+)	(25, 25, 25, 25, 30, 31+)
(13, 13, 13, 13, 15, 16+)	(19, 19, 19, 19, 23, 24+)	(25, 25, 25, 25, 31, 32+)
(13, 15, 15, 15, 16, 17+)	(19, 21, 21, 21, 24, 25+)	(25, 27, 27, 27, 32, 33+)
(13, 15, 15, 15, 17, 18+)	(19, 21, 21, 21, 25, 26+)	(25, 27, 27, 27, 33, 34+)
(13, 17, 17, 17, 18, 19+)	(19, 23, 23, 23, 26, 27+)	(25, 29, 29, 29, 34, 35+)

Note that the second and last families are defined only for $x_1=4m+3$ and $x_1=6m+1$, respectively, where $m\ge 3$. This complicated pattern shows that it is hardly possible to combine all exceptions by a formula.

2.9. Odd $x_1\ge 3$ with $n={\displaystyle \frac{1}{2}}(x_1+1)+2$

For $n-{\displaystyle \frac{1}{2}}(x_1+1)>2$, no exceptions were found, while in the considered case, the exceptions are as follows. Given an integer $i\ge 0$, a position $x=(x_1,\dots ,x_n)$ is an exception if and only if

$$x_1+i=x_2=\dots =x_{n-1}<x_n\le \dots \le x_m,i=0,1,\dots$$

Furthermore, the remoteness function is given by the formula $\mathcal{R}\left(x\right)=f(x_1+i+1)$, where function f was defined in Section 2.3.

Note that $x_2=\dots =x_{n-1}$ and this number is even if and only if i is odd.

Examples for $x_1=3,5,7,9,11$ and, respectively, $n=4,5,6,7,8$ are given below.

$x_1=3,n=4$	$\mathcal{R}$	$x_1=5,n=5$	$\mathcal{R}$	$x_1=7,n=6$	$\mathcal{R}$
(3, 3, 3, 4+)	5	(5, 5, 5, 5, 6+)	7	(7, 7, 7, 7, 7, 8+)	9
(3, 4, 4, 5+)	5	(5, 6, 6, 6, 7+)	7	(7, 8, 8, 8, 8, 9+)	9
(3, 5, 5, 6+)	7	(5, 7, 7, 7, 8+)	9	(7, 9, 9, 9, 9, 10+)	11
(3, 6, 6, 7+)	7	(5, 8, 8, 8, 9+)	9	(7, 10, 10, 10, 10, 11+)	11
(3, $\stackrel{.}{7}$, 7, 8+)	9	(5, 9, 9, 9, 10+)	11	(7, 11, 11, 11, 11, 12+)	13

$x_1=9,n=7$	$\mathcal{R}$	$x_1=11,n=8$	$\mathcal{R}$
(9, 9, 9, 9, 9, 9, 10+)	11	(11, 11, 11, 11, 11, 11, 11, 12+)	13
(9, 10, 10, 10, 10, 10, 11+)	11	(11, 12, 12, 12, 12, 12, 12, 13+)	13
(9, 11, 11, 11, 11, 11, 12+)	13	(11, 13, 13, 13, 13, 13, 13, 14+)	15
(9, 12, 12, 12, 12, 12, 13+)	13	(11, 14, 14, 14, 14, 14, 14, 15+)	15
(9, 13, 13, 13, 13, 13, 14+)	15	(11, 15, 15, 15, 15, 15, 15, 16+)	17

2.10. Odd $x_1\ge 5$ with $n={\displaystyle \frac{1}{2}}(x_1+1)+1$

Given an integer $i\ge 0$, a position $x=(x_1,\dots ,x_n,\dots ,x_m)$ is an exception if and only if one of the following two cases holds:

$$x_1,x_2=x_1+2i,x_3=\cdots =x_{n-1}=x_2+1<x_n\le \cdots \le x_m,$$

$$x_1,x_2=x_1+2i,x_3=\cdots =x_{n-1}=x_2+2<x_n\le \cdots \le x_m.$$

Furthermore, the remoteness function is given by formula $\mathcal{R}\left(x\right)=f\left(x_n\right)$, where function f is defined above. Again, it is easily seen that $\mathcal{R}\left(x\right)$ is odd for each i; hence, the first player wins in every exceptional position, but (s)he loses if (s)he follows the M-rule.

Finally, the unique optimal move in x is to keep $x_n$ when $x_{n-1}$ is even and $x_{n-1}$ when it is odd. In contrast, the unique M-move in x is to keep $x_n$ when $x_{n-1}$ is odd and $x_{n-1}$ when it is even. Thus, the sets of optimal moves and M-moves are disjoint in every exceptional position.

Note that $x_3=\dots =x_{n-1}$ and this number is even if and only if i is odd.

Examples for $x_1=5,7,9,11,13$ and, respectively, $n=4,5,6,7,8$ are given below.

$x_1=5,n=4$	$\mathcal{R}$	$x_1=7,n=5$	$\mathcal{R}$	$x_1=9,n=6$	$\mathcal{R}$
(5, 5, 6, 7+)	7	(7, 7,8, 8, 9+)	9	(9, 9, 10, 10, 10, 11+)	11
(5, 5, 7, 8+)	9	(7, 7, 9, 9, 10+)	11	(9, 9, 11, 11, 11, 12+)	13
(5, 7, 8, 9+)	9	(7, 9, 10, 10, 11+)	11	(9, 11, 12, 12, 12, 13+)	13
(5, 7, 9, 10+)	11	(7, 9, 11, 11, 12+)	13	(9, 11, 13, 13, 13, 14+)	15
(5, 9, 10, 11+)	11	(7, 11, 12, 12, 13+)	13	(9, 13, 14, 14, 14, 15+)	15

$x_1=11,n=7$	$\mathcal{R}$	$x_1=13,n=8$	$\mathcal{R}$
(11, 11, 12, 12, 12, 12, 13+)	13	(13, 13, 14, 14, 14, 14, 14, 15+)	15
(11, 11, 13, 13, 13, 13, 14+)	15	(13, 13, 15, 15, 15, 15, 15, 16+)	17
(11, 13, 14, 14, 14, 14, 15+)	15	(13, 15, 16, 16, 16, 16, 16, 17+)	17
(11, 13, 15, 15, 15, 15, 16+)	17	(13, 15, 17, 17, 17, 17, 17, 18+)	19
(11, 15, 16, 16, 16, 16, 17+)	17	(13, 17, 18, 18, 18, 18, 18, 19+)	19

2.11. Odd $x_1\ge 7$ with $n={\displaystyle \frac{1}{2}}(x_1+1)$

Given an integer $i\ge 0$, a position $x=(x_1,\dots ,x_n,\dots ,x_m)$ is an exception if and only if one of the following two cases holds:

$$x_1,x_2=x_1+2i,x_3=\dots =x_{n-1}=x_2+3<x_n\le \dots \le x_m,$$

$$x_1,x_2=x_1+2i,x_3=\dots =x_{n-1}=x_2+4<x_n\le \dots \le x_m.$$

Interestingly, all further arguments can be copied from the previous subsection without any changes; however, we should remember that n is reduced by 1.

Examples for $x_1=7,9,11,13,15$ and, respectively, $n=4,5,6,7,8$ follow.

$x_1=7,n=4$	$\mathcal{R}$	$x_1=9,n=5$	$\mathcal{R}$	$x_1=11,n=6$	$\mathcal{R}$
(7, 7, 10, 11+)	11	(9, 9, 12, 12, 13+)	13	(11, 11, 14, 14, 14, 15+)	15
(7, 7, 11, 12+)	13	(9, 9, 13, 13, 14+)	15	(11, 11, 15, 15, 15, 16+)	17
(7, 9, 12, 13+)	13	(9, 11, 14, 14, 15+)	15	(11, 13, 16, 16, 16, 17+)	17
(7, 9, 13, 14+)	15	(9, 11, 15, 15, 16+)	17	(11, 13, 17, 17, 17, 18+)	19
(7, 11, 14, 15+)	15	(9, 13, 16, 16, 17+)	17	(11, 15, 18, 18, 18, 19+)	19

$x_1=13,n=7$	$\mathcal{R}$	$x_1=15,n=8$	$\mathcal{R}$
(13, 13, 16, 16, 16, 16, 17+)	17	(15, 15, 18, 18, 18, 18, 18, 19+)	19
(13, 13, 17, 17, 17, 17, 18+)	19	(15, 15, 19, 19, 19, 19, 19, 20+)	21
(13, 15, 18, 18, 18, 18, 19+)	19	(15, 17, 20, 20, 20, 20, 20, 21+)	21
(13, 15, 19, 19, 19, 19, 20+)	21	(15, 17, 21, 21, 21, 21, 21, 22+)	23
(13, 17, 20, 20, 20, 20, 21+)	21	(15, 19, 22, 22, 22, 22, 22, 23+)	23

There exists another family of exceptions for $x_1\ge 9$ with $n={\displaystyle \frac{1}{2}}(x_1+1)$.

$$x_1,x_2=x_3=x_1+2i,x_4=\dots =x_{n-1}=x_2+3<x_n\le \dots \le x_m,$$

$$x_1,x_2=x_3=x_1+2i,x_4=\dots =x_{n-1}=x_2+4<x_n\le \dots \le x_m.$$

Examples for $x_1=9,11,13$ and, respectively, $n=5,6,7$ follow.

$x_1=9,n=5$	$\mathcal{R}$	$x_1=11,n=6$	$\mathcal{R}$	$x_1=13,n=7$	$\mathcal{R}$
(9, 9, 9, 10, 11+)	11	(11, 11, 11, 12, 12, 13+)	13	(13, 13, 13, 14, 14, 14, 15+)	15
(9, 9, 9, 11, 12+)	13	(11, 11, 11, 13, 13, 14+)	15	(13, 13, 13, 15, 15, 15, 16+)	17
(9, 11, 11, 12, 13+)	13	(11, 13, 13, 14, 14, 15+)	15	(13, 15, 15, 16, 16, 16, 17+)	17
(9, 11, 11, 13, 14+)	15	(11, 13, 13, 15, 15, 16+)	17	(13, 15, 15, 17, 17, 17, 18+)	19
(9, 13, 13, 14, 15+)	15	(11, 15, 15, 16, 16, 17+)	17	(13, 17, 17, 18, 18, 18, 19+)	19

2.12. Odd $x_1\ge 9$ with $n={\displaystyle \frac{1}{2}}(x_1+1)-1$

Exceptions for $x_1=9,11,13,15$ and, respectively, $n=4,5,6,7$ follow:

$x_1=9,n=4$	$\mathcal{R}$	$x_1=11,n=5$	$\mathcal{R}$	$x_1=13,n=6$	$\mathcal{R}$
(9, 9, 14, 15+)	15	(11, 11, 13, 14, 15+)	15	(13, 13, 13, 13, 14, 15+)	15
(9, 9, 15, 16+)	17	(11, 11, 13, 15, 16+)	17	(13, 13, 13, 13, 15, 16+)	17
(9, 11, 16, 17+)	17	(11, 13, 15, 16, 17+)	17	(13, 15, 15, 15, 16, 17+)	17
(9, 11, 17, 18+)	19	(11, 13, 15, 17, 18+)	19	(13, 15, 15, 15, 17, 18+)	19
(9, 13, 18, 19+)	19	(11, 15, 17, 18, 19+)	19	(13, 17, 17, 17, 18, 19+)	19

$x_1=13,n=6$	$\mathcal{R}$	$x_1=15,n=7$	$\mathcal{R}$
(13, 13, 15, 16, 16, 17+)	17	(15, 15, 15, 15, 16, 16, 17+)	17
(13, 13, 15, 17, 17, 18+)	19	(15, 15, 15, 15, 17, 17, 18+)	19
(13, 15, 17, 18, 18, 19+)	19	(15, 17, 17, 17, 18, 18, 19+)	19
(13, 15, 17, 19, 19, 20+)	21	(15, 17, 17, 17, 19, 19, 20+)	21
(13, 17, 19, 20, 20, 21+)	21	(15, 19, 19, 19, 20, 20, 21+)	21

2.13. Odd $x_1\ge 11$ with $n={\displaystyle \frac{1}{2}}(x_1+1)-2$

For $n=(x_1+1)/2-2$, we obtained the following exceptions:

$x_1=11,n=4$	$\mathcal{R}$	$x_1=13,n=5$	$\mathcal{R}$	$x_1=15,n=6$
(11, 11, 18, 19+)	19	(13, 13, 20, 20, 21+)	21	(15, 15, 22, 22, 22, 23+)
(11, 11, 19, 20+)	21	(13, 13, 21, 21, 22+)	23	(15, 17, 23, 23, 23, 24+)
(11, 13, 20, 21+)	21	(13, 15, 22, 22, 23+)	23	(15, 17, 24, 24, 24, 25+)
(11, 13, 21, 22+)	23	(13, 15, 23, 23, 24+)	25	(15, 19, 25, 25, 25, 26+)
(11, 15, 22, 23+)	23	(13, 17, 24, 24, 25+)	25	(15, 19, 26, 26, 26, 27+)

In addition, the following exceptions were found:

$x_1=13,n=5$	$\mathcal{R}$		$x_1=15,n=6$		$x_1=17,n=7$
(13, 13, 13, 16, 17+)	17		(15, 15, 23, 23, 23, 24+)		(17, 17, 17, 17, 17, 18, 19+)
(13, 13, 13, 17, 18+)	19		(15, 17, 24, 24, 24, 25+)		(17, 17, 17, 17, 17, 19, 20+)
(13, 15, 15, 18, 19+)	19		(15, 17, 25, 25, 25, 26+)
(13, 15, 15, 19, 20+)	21		(15, 19, 26, 26, 26, 27+)
(13, 17, 17, 20, 21+)	21		(15, 19, 27, 27, 27, 28+)

2.14. Odd $x_1\ge 13$ with $n={\displaystyle \frac{1}{2}}(x_1+1)-3$

Exceptions for $x_1=13,15,17$ and, respectively, $n=4,5,6$ follow:

$x_1=13,n=4$	$x_1=15,n=5$	$x_1=17,n=6$
(13, 13, 22, 23+)	(15, 15, 24, 24, 25+)	(17, 17, 26, 26, 26, 27+)
(13, 13, 23, 24+)	(15, 15, 25, 25, 26+)	(17, 17, 27, 27, 27, 28+)
(13, 15, 24, 25+)	(15, 17, 26, 26, 27+)	(17, 19, 28, 28, 28, 29+)
(13, 15, 25, 26+)	(15, 17, 27, 27, 28+)	(17, 19, 29, 29, 29, 30+)
(13, 17, 26, 27+)	(15, 19, 28, 28, 29+)	(17, 21, 30, 30, 30, 31+)