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1. Introduction

In this article we study new (bitter) chocolate games.  A.C.Robin introduced a rectangle chocolate game in [1] , but it was equivalent to the traditional game of nim.
The authors have presented new chocolate games whose mathematical structures were different from that of traditional game of nim in [5] , [6] and [3] . The coordinates of the game in [5] and [4] satisfy the inequality , and the coordinates of the game in [6] satisfy the inequality . As to the coordinates of chocolate see Definition 1.4 of this article.
In the authors presented many kinds of graphs produced by the set of L-states of the chocolate games.
In Section 1 the authors present new chocolate games, and the coordinates of the games satisfy the inequality . In this section a formula for the set of L states of this chocolate game is presented.
In Section 2 we prove this formula. This formula has been already presented in [7], but the essential part of the proof that is presented in this section is more simple and clearer than that in [7].
In [7] the authors used Lemma 2.3 (of [7]) to prove Theorem 2.2 (of [7]), but many people told authors that this Lemma and this Theorem are difficult to understand. In this article authors make Lemma 2.4 to prove Theorem 2.2 that is mathematically equivalent to Theorem 2.2 in [7]. By using Lemma 2.4 the proof of Theorem 2.2 becomes quite clear, and this is one of the important points of this article.

In Section 3 the authors presented two predictions for the case of . They have not managed to prove these, but they present the result of the calculation by Mathematica ( a computer algebra system) to show that these predictions seem to be plausible.
In Section 4 the authors present some interesting graphs made by the sets of L states of the chocolate for . In this section the authors do not present any theory of mathematics, but the graphs presented here are very unique and they cannot be found anywhere else.

First, we define the chocolate games. See the example of chocolates in Example 1.1.

Definition 1.1   Given a pieces of chocolate, where the light green parts are sweet and the red part is very bitter. This game is played by two players in turn. Each player breaks the chocolate (in a straight line along the grooves) and eats the piece he breaks off. The player to leave his opponent with the single bitter part is the winner.

Remark 1.1   Note that in Definition 1.1 we are not considering a misere play, since the player who breaks the chocolate for the last time is the winner, since his opponent cannot break the remaining red (bitter) polygon. Therefore the chocolate games in this article are normal play games.

Example 1.1   We study the chocolate in Fig 1.1.   We call this chocolate as a chocolate by using the number of rows and columns of the chocolate. We cut chocolate along the black lines, and the chocolates can be cut horizontally or vertically. When a player cut the chocolate into two parts, he eat the part that does not contain the bitter part of the chocolate. If he cut the chocolate vertically, he leaves the chocolate. For example if he cut one column, then he leaves the chocolate of Fig1.2. If he cut the chocolate horizontally, he leaves the chocolate. For example if he cut one row or two rows, then he leaves the chocolate of Fig1.2 or Fig 1.4 respectively.
If we start with the chocolate in Fig 1.1. , any chocolate that appears in the game can be expressed as a chocolate.

Figure 1.1  

Figure 1.2  

Figure 1.3  

Figure 1.4  

Suppose that we play with the chocolate in  Fig 1.1 , then which player wins the game the first or the second?  In fact the first player can win the game.  The method is quite simple.  If you play as the first player, you have only to move to Fig 1.4.  Then your opponent will go to a chocolate or a chocolate.  Then you have only to go to a or chocolate.  The strategy is go to a square chocolate every time, and finally you reach the chocolate and win the game.

Definition 1.2   Here we define two important states of chocolates.
W States, from which we can force a win, as long as we play correctly at every stage.
L States, from which we will lose however well we play, but we may end up winning if our opponents make a mistake.

Remark 1.2   It is important to realize that the outcome of this game is not pre-determined, but there is nothing that the potential loser can do if the potential winner plays correctly at ever stage. However the potential winner cannot afford to make a single mistake, for if he does, then his opponent can exploit this.

One of the most important topics of chocolate games is to find all the L states and the W states of games. We denote by the set of non-negative integers.

Remark 1.3   For the L states of the chocolate of Fig 1.1 see the  corner in  [1] . See also Theorem 7.12 in [2] (p-138). Note that in [2] they use the term "P-position" instead of "L state" of this article.

We define nim-sum that is important for the theory of games.

Definition 1.3   Let be non-negative integers, and write them in base , so and with .
We define the nim-sum by


where .

Theorem 1.1   Let .Then
If , and ,then
we have .

This is a well known result of nim-sum. For a proof see Theorem 7.12 in [2] (p-139).

Next we study the chocolates in Fig 1.5 that is a new chocolate game that was made by the authors. 


      Figure 1.5  

We generalize the chocolate games in Fig 1.5, and define one of the chocolate games that we study in this article.

Definition 1.4   We make a triangle on the 2 dimensional plane with two coordinates by connecting three points , , with segments, where and is a natural number.
We connect points and with segments for , and we also connect points and with segments for , where . We make a trapezoid with four points , , , , and color this trapezoid with red. We color all the other parts of the triangle with green. All the segments are in black.
green polygons are sweet chocolate that can be eaten, and the red trapezoid is the bitter chocolate that cannot be eaten.

                                  Figure 1.6  

You can cut these chocolate along the segments in three ways.
You cut parallel to the segment on the left side of the red(bitter) trapezoid.
You cut horizontally above the red (bitter) trapezoid.
You cut parallel to the segment on the right side of the red trapezoid.
Therefore it is proper to represent these chocolates with , where x,y,z stand for the maximum numbers of times that we can cut these chocolate in each direction. For example in Fig 1.5 we can cut 2 times at most parallel to the segment on the left side of the red trapezoid, 2 times at most horizontally and 5 times at most parallel to the segment on the right side of the red trapezoid. Therefore , and . Therefore we represent the chocolate in Fig 1.5 with the coordinates .

For any we denote by the state of coordinates.

Example 1.2   Here we have four examples of states of chocolates that appear when we play the chocolate game of Fig 1.5

Figure 1.7  

Figure 1.8  

Figure 1.9  

Figure 1.10  

It is clear that the coordinates of these states satisfy the inequality , and this is equivalent to the inequality


where is the floor function. Note that is the largest integer not greater than for any real number . Inequality (1.2) is important to understand the structure of the chocolate. If you start with the chocolate in Fig 1.5 and reduce the third coordinate to by cutting from the upper left to the lower right on the right side of the bitter polygon, then by the structure of the chocolate the second coordinate is reduced to .
In this way we get the chocolate in Fig 1.7.
If we are to explain the move from the chocolate in Fig 1.5 to the chocolate in Fig 1.7 using Inequality (1.2), we use the following explanation.
Let , then by Inequality (1.2) we get .

Definition 1.5   Let and , and .

Theorem 1.2   Let and be the sets defined in Definition 1.5. is the set of L states and is the set of W states of the game of Definition 1.4.

By Theorem 1.2 we tell L states from W states, and in Example 1.3 we learn how to win the game using the theorem.

We prove this theorem in Section 2.
First, we learn how to use the theorem.

Example 1.3   Here we use Theorem 1.2, then the chocolate in Fig 1.5 with the coordinates is a W state, since . Therefore you can win if you play as the first player. The strategy is to move to an L state every time. For example you can move to that is an L state, since .
In this way you should leave the opponent with an L state every time until you leave the opponent with the single block of bitter chocolate whose coordinates are . Then the game is finished, and you win the game.

By Example 1.3 the strategy to win is clear. Even if you do not read the proof of Theorem 1.2, you can be a good player of this chocolate game.

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