In Subsection 3.1, we proved that the Grundy number for when the function satisfies the condition in Definition 3.2.

In this subsection, we prove that the condition in Definition 3.2 is a necessary condition for to have the Grundy number for the chocolate bar . Let be a monotonically increasing function of into that satisfies the following condition .

Suppose that is the Grundy number of the chocolate bar . Then,

Throughout this subsection we assume that the function satisfies the condition of Definition 3.4, and we prove that this function satisfies the condition of Definition 3.2 using the following Lemma 3.9, Lemma 3.10 and Lemma 3.11.

Lemma 3.9. Let such that , and . Then, .

Proof. Since , we have for . Therefore,

, where .

(3.44) |

Since , . Therefore, (3.44) implies .

Lemma 3.10. For any such that , we have . Let such that , and let

(3.45) |

Then, Equation (3.45) implies that . By the definition of Grundy number, there exist such that and .

By the definition of , we have the following Equation (3.46) or Equation (3.47).

(3.46) |

for with .

(3.47) |

for with . Equation (3.47) contradicts Equation (3.45), and hence we have Equation (3.46).

If

(3.48) |

then . Hence,

(3.49) |

By Equations (3.46) and (3.49), we have

(3.50) |

Since , Equation (3.50) leads to a contradiction. Therefore, the inequality in (3.48) is false, and we have . Hence, Equation (3.46) implies that . The number of elements in is the same as the number of elements in , and hence we have .

Lemma 3.11 Let

for , , and .

If for , then . Let

(3.51) |

for . We suppose that

(3.52) |

and we show that this leads to a contradiction.

Case If , then

(3.53) |

By Equation (3.51), the inequality in (3.52) and Lemma 3.9, we have , which contradicts Equation (3.53).

Case If , then

. Note that , since and .

Therefore, by the definition of Grundy number we have

(3.54) |

(3.55) |

Note that .

For , we have . Hence

(3.55) | |||

(3.56) |

Since , Lemma 3.10 implies that . Therefore, by Definition 3.4

(3.56) | |||

(3.57) | |||

(3.58) | |||

(3.59) | |||

(3.60) |

Then we have the following statements , , and .

All the numbers in Set (3.57) are of the type , and hence this set does not contains . Note that .

The coefficients of of the numbers in Set (3.58) are not , and hence this set does not contains .

All the numbers in Set (3.59) are of the type , and hence this set does not contains .

Note that and .

The coefficients of of the numbers in Set (3.60) are not , and hence this set does not contains .

Statements , , and contradict Relation (3.54). Therefore we conclude that the inequality in (3.52) is false.

Theorem 3.3. Suppose that the function satisfies the condition in Definition 3.4.

Then function satisfies the condition in Definition 3.2.

Proof. If does not satisfy the condition in Definition 3.2, then there exist and a natural number such that ,

(3.61) |

and

(3.62) |

By Equation (3.61), there exist for such that and . By the inequality in (3.62), there exist for and a natural number such that ,

(3.63) |

and

(3.64) |

Let . Then . Hence Equations (3.63) and (3.64) imply that

(3.65) |

and

(3.66) |

Let

(3.67) |

and

(3.68) |

Then, follows directly from (3.67) and (3.68). Let . Then, . By (3.65),

, and hence Equation (3.67) implies

(3.69) |

By , we have .

Hence Equations (3.68), (3.66) and the inequality in (3.69) implies that . Therefore, there exist and such that , and .

By Equation (3.65) and the inequality in (3.69), we have

(3.70) |

Equation (3,67) implies

(3.71) |

and

(3.72) |

Inequalities (3.70), (3.71) and (3.72) contradict the result of Lemma 3.11, and hence that satisfies the condition in Definition 3.1. By Theorem 3.3, the condition in Definition 3.1 is a necessary condition for to have the Grundy number .