## 1 Introduction  
In this paper we analyze further the _Magnus-Derek_ game, a two-player game introduced in [1]. The game is played on a circular table with \(n\) positions labeled consecutively from \(0\) to \(n-1\) in a clockwise direction. A single token is initially placed at position \(0\). A round consists of the first player, [PERSON] (from _magnitude_) choosing an integer magnitude \(0<\ell\leq\frac{n}{2}\), followed by the second player, [PERSON] (from _direction_) choosing whether the token will move clockwise or counter-clockwise. The token is then moved \(\ell\) positions in the chosen direction. [PERSON]'s primary goal is to maximize the total number of distinct positions visited by the token over the course of the game (and secondarily, to minimize the number of rounds), while [PERSON] main aim is to minimize the number of positions visited. These goals are natural since for all \(n\) not exact power of \(2\), [PERSON] has a strategy to prevent the token from visiting some of the positions [1]. For brevity, we will skip from now the word distinct in all expressions of type \"maximize the total number of distinct positions visited.\"  
In our analysis we will use \(\mathbb{Z}_{n}=\{0,1,2,\ldots,n-1\}\) to represent the table of size \(n\). Let \(K\in\mathbb{Z}_{n}\) denote the position of the token at the beginning of a round. The round then consists of [PERSON] choosing a magnitude \(0<\ell\leq\frac{n}{2}\), [PERSON] choosing a direction \(+\) or \(-\) (where the positive direction corresponds to \"clockwise\"), and the token moving to \((K\pm\ell)\mod n\).  
In [1,2], the following function on the set of positive integers was used. Let \(n>0\) be an integer. Then  
\[f^{*}(n)=\left\{\begin{array}{ll}n&\mbox{if $n=2^{k}$ for some $k>0$,}\\ \\ \frac{p-1}{p}\cdot n&\mbox{otherwise, where $p$ is the smallest odd prime factor of $n$.}\end{array}\right.\]  
In [1] (and subsequently with a shorter proof in [2]), the following theorem was obtained.  
**Theorem 1**.: \(\bullet\) _[PERSON] has a strategy to prevent the token from visiting more than \(f^{*}(n)\) positions no matter how long the game is played._  
\(\bullet\) _[PERSON] has a strategy to ensure the token visits at least \(f^{*}(n)\) positions._  
One of the strategies previously given for [PERSON] by [PERSON] and [PERSON] requires \(O(n^{2})\) rounds [1]. A recent paper by [PERSON], [PERSON], and [PERSON] gives an \(O(n\log(n))\)-round strategy [3]. In this paper, we will show that [PERSON] can visit \(f^{*}(n)\) positions in \(3(n-1)\) rounds, the optimal number of rounds up to a constant factor. Although some of the basic ideas in the two papers are the same, we wrote our paper well before the paper of [PERSON], [PERSON], and [PERSON] was published [4], and our results were found independently from theirs.  
Our strategy for [PERSON] requires that he know the factorization of \(n\). To our knowledge, this is the first game in which the ability to factor conveys an advantage to a player. In another variant of the game called the Vector game [5], the players' goals are reversed: [PERSON] wants to minimize the set of occupied positions, while [PERSON] aims to maximize the number of positions visited. The ability to factor can also give a crucial advantage to a player here. Moreover the analysis of this second variant of the game is also not completely done yet; it created several new open problems (see [5,6]).  
As mentioned in [1], the sequence \(\{f^{*}(n)\}_{n=1}^{\infty}=(1,2,2,4,4,4,6,8,6,8,\ldots)\) diverges non-monotonically and non-linearly to \(\infty\). This sequence has not previously appeared in the Encyclopedia of Integer Sequences [7].