Generalizations of the Feline and Texas Chainsaw Josephus Problems

We define and study the Extended Feline Josephus Game, a game in which n players, each with l lives, stand in a circle. The game proceeds by alternating between hitting k consecutive players—each of whom will consequently lose a life—and skipping s consecutive players. This cycle continues until every player except one loses all of their lives. Given the nonnegative integer parameters n, k, s and l, the goal of the game is to identify the surviving player. In this paper, we show how the defining parameters n, k, s, and l affect the survivor of games with specific constraints on those parameters and our main results provide new closed formulas to determine the survivor of these Extended Feline Josephus Games. Moreover, for cases where these formulas do not apply, we provide recursive formulas for reducing the initial game to other games with smaller parameter values. For the interested reader, we present a variety of directions for future work in this area, including an extension which considers players lying on a general graph, rather than on a circle.

come of the game, Josephus and one other man remained until the final round of the game, at which point the two decided to surrender to the Roman forces [1] [2].
This historical account led mathematicians to define and study variations of the counting-out game, named the Josephus Game, which is described as follows: Given an ordered set of n players standing in a circle, the game begins by determining an initial player from which we alternate between skipping and killing a player in a clockwise direction. The Josephus Problem is to identify the position of the surviving player in this game. Although the problem has been solved in various ways [3] [4] [5] [6], many variations and generalizations of the game have since been introduced.
Two of these generalizations are the Generalized Josephus Game, a variant that allows the number of consecutively skipped players to equal some fixed positive integer [2] [4] [5] [7], and the Feline Josephus Game [2] [8], which adds the rule that each player now has a positive integer number of lives. In the Feline Josephus Game, each time a player is hit; their number of lives decreases by 1. If a player's number of lives reaches 0, that player is removed from the game, and the game continues until all but one player remains. Other variants of the game include the Texas Chainsaw and Feline Texas Chainsaw Josephus Games. In these variants, the rules of the original Josephus Game still apply; however the number of players killed in a row varies instead of the number of players skipped. As the name implies, the latter includes the varying lives aspect from the Feline variant [2] [9].
In this work, we introduce the Extended Feline Josephus Game with the goal of identifying the survivor when the number of players, consecutive hits, consecutive skips, and initial lives, are greater than or equal to 1. We remark that this new variant recovers the Feline variant when setting the hit parameter to 1, the Feline Texas Chainsaw variant when setting the skip parameter to 1, the Texas Chainsaw variant when setting both the skip and lives parameters to 1, and the original Josephus Game when setting the hit, skip, and lives parameters to 1. Thus the Extended Feline Josephus Game we present provides a generalization to much of the previous work in this area.
The main results in this paper determine closed formulas and recurrence relations for the survivor of an Extended Feline Josephus Game when the defining parameters satisfy certain relations. In what follows, we let ( ) / , , , K S J n k s  denote the surviving player of the Extended Josephus Game given n people, k hits, s skips, ℓ lives, and where the subscript denotes whether the game begins by hitting or skipping first. If both of the subscripts K and S are present, this means that the result works for each of these cases. Lastly, we remark that our games always label players 0 through n and they begin at player 0.
Our first main result presents the case where 1 =  and provides closed formulas to determine the survivor of the games considered.
The paper is organized as follows. In Section 2, we give the necessary background and notation to concretely define the Extended Feline Josephus Game.
In Section 3, we consider explicit solutions for games where the lives parameter is 1, and present a proof of Main Theorem 1. In Section 4, we consider closed formulas to games where the lives parameter varies. In Section 5, we consider recursive solutions for reducing games to other games with smaller defining parameters thereby establishing Main Theorem 2. Lastly, Section 6 communicates ideas and areas for future work.

Background
To understand the Extended Feline Josephus Game, we begin by defining the variables and notations used to describe the layout for any particular game. We label the n players 0 through creases by 1, and a player is removed from the game once their number of lives reaches 0. In these games, k consecutive players are hit in a row, and then s consecutive players are skipped. The last variable, p, denotes the player whom the game begins with. Whether the game starts by skipping or by killing players first is referred to as either a skip-first or kill-first game, and this is denoted through the use of subscripts S or K, respectively. Thus for a specific game ( ) / , , , , K S n k s p  , the surviving player is denoted as ( ) / , , , , K S J n k s p  .
We also define a round as one set of k hits and s skips (or vice versa) and a cycle as a traversal through all (remaining) players in a game regardless of whether the action on any specific player is a hit or a skip.
We illustrate these definitions through the following example 1 . Example 1. We want to determine the survivor of a kill-first game where 10 n = , , and 0 p = . We run through the game in Figure  1, by denoting a hit with an X and a skip with an O. Whenever a player has lost all of their lives we remove them from the game, in which case they receive neither an X nor an O for the remaining rounds. This allows us to determine that

( )
The following remark and result demonstrate that any game can be translated into another with a different starting player or to a kill-first/skip-first order while preserving the final position of the survivor. As a result of Remark 1, all following results utilize games beginning at player 0. Therefore, for the sake of simplicity, from now on we denote the survivor of the game as ( ) / , , , K S J n k s  , eliminating the need for the p parameter. Also, the use of kill-first or skip-first games in describing specific results depending on which is more intuitive to understand.

Games Where Players Have One Life
We begin our analysis by establishing results for the Extended Feline Josephus Game in cases where 1 =  . We will use these results to extend to more general The following result provides an initial identification of the survivor of a game under these parameters.
Proof. Since n k ≤ , the life of every player is decreased by 1 in the first round.
, each player's life is decreased to 0, and each player is removed from the game. Therefore, the survivor is the last player to be removed, player We then shift the players positions forward by k to return to the original game layout. Thus, at the end of the first round, the last player skipped is at position , the number of remaining players is strictly less than half of the initial number of players at the beginning of the game and thus less than or equal to k. By Proposition 1, the second round proceeds by removing all the remaining players starting at the player mod k a n + and ending at position 1mod k a n + − . Therefore, the survivor is player 1mod k a n + − .  Having established the prior results, we now use a proof technique similar to Proof. The proof consists of three cases based on the value of n. We first consider case 1 where n k ≤ . Note that a n ≡ , so 0 b = and 0 m = . Thus by Proposition 1 and Lemma 1, the survivor is player ( ) Next, we consider case 2 where n k > and b k s n a s If n k s = + , then by Proposition 2 and Lemma 1, the survivor is and thus 2 . Therefore by Proposition 3 and Lemma 1, the survivor is we know that there are exactly n k s + rounds in a cycle through all of the players.
Since in each round, we skip s out of the k s + players in the round, it follows that at the end of the first cycle, there are players remaining, the next round starting at player 0. Similarly, after a total of b cycles, there are only a players remaining since b is reduced to 0. Note that in each cycle, players 0 through 1 s − are skipped and are thus part of the remaining a players. Since a k ≤ , and the next round starts at player 0, this case reduces back to case 1. Thus, the survivor is player ( ) Finally, we consider case 3 where n k > and b k s n a km s

Games Where Players Have Multiple Lives
In order to find the surviving player in games where 1 ≥  , we first consider closed formulas of special cases with respect to n. Proof. Since n k ≤  , the lives of every player is decreased to 0 in the first round, removing every player. The survivor is player 1 n − , the last to be removed.
 Similarly to Proposition 1, the more interesting cases are where n k >  , which the following formulas explore. The results are derived from attempts to find solutions to the cases where n k = and n s = , respectively. s n a a n n a n J n k s n a n a n Proof. Since | n s , by Lemma 1, the surviving player of a skip-first game of this form is the same as its kill-first counterpart. It suffices to only consider the skip-first game.
We first look at the case where a n =  . Since a n =  , we know that | k n and that n k  is an integer. It follows that n k  is the total number rounds in the game, and since s is a multiple of n, every player is hit in order and thus also removed in order in the last round. Therefore, the survivor is player 1 n − .
Since k n >  , by Proposition 4, the survivor is at position 1 n − .  To better illustrate the proof of Proposition 6, we provide the following.

Reducing Parameter Values of Games
If the problem does not fall into one of the cases where there is a known closed formula for identifying the survivor, the best alternative is to reduce the amount of time it takes to identify the survivor iteratively. We do this by reducing the parameters involved with a particular game.
Note that for certain games where parameters n, k, and s are fixed, we see a repeating sequence of numbers for the survivor as ℓ increases. In order to more clearly illustrate, we define an step as any act occurring in the game, whether it be a skip, hit, or kill. Additionally we define a reset as a point in a game where after a number of rounds, the next round begins at the initial player, and every player in the game has had their lives decreased by the same amount. The game has effectively been simplified to a game starting at the same player with the same layout, only with fewer lives.
Proof. Since Example 5. We want to know the survivor of a skip-first game where 11 n = , 6 k = , 5 s = , and 3 =  . We run the game in Figure 4.
Note that the layout of the game now has s players. Since k s ≥ , by Proposition 6, the survivor is the same.  Example 6. We want to know the survivor of a skip-first game where 11 n = , 6 k = , 5 s = , and 3 =  . Doing so will require the use of Example 3. We run the game in Figure 5.   The following example illustrates the proof of Theorem 8.  Example 7. We want to know the survivor of the skip-first game with 15 n = , 2 k = , 3 s = , and 2 =  . We run the example in Figure 6.
Using Theorem 8, we calculate that 3 a = and find that ( ) 3  Note that while k and s are fixed parameters, there values can still be reduced.
In particular, their values can be reduced as long as they retain the same proportions relative to n. The following result, offers an equation to recover the initial game's survivor using this method of reduction. This is also the content of Part 5 of Main Theorem 2.

Future Work
In this paper, we showed how parameters , , n k s , and ℓ affect the survivor of games with specific constraints on those parameters. However, the general problem of identifying the survivor with any set of parameters has not yet been solved. Similar to Sullivan and Inkso's variant, it is of note that since ℓ can be reduced to less than or equal to k in any game where n and k s + are relatively prime, any direct solution for this case would only need to consider k ≤  .
Other questions include but are not limited to: Are there other relationships between n, k, and s which produce similar reduction patterns? For many of the results in this paper, k s ≥ . Through working with this problem, we noticed that the survivor becomes harder to identify as k becomes smaller, relative to s or n. Is there a recursive solution for cases when k is relatively close to either s or n?
Another direction for further research includes exploring other Josephus game variants. For example, S. Sharma, Tripathi, Bagai, Saini, and N. Sharma explore a variant in which the number of people hit at a time varies with each round [11]. One could then study a game variant in which the number of skips and hits vary within the game. Lastly, in the game studied here, the n players are listed in a line/circle. A further extension can be considering them lying on a general graph. Therefore, one could apply the techniques of random walks to study such games. For relevant work we refer the reader to [12].