4.1 A Sufficient Condition for a Chocolate Bar to have the Grundy Number

Let be the function that satisfies the condition in Definition 3.2 and let be the Grundy number of . The condition in Definition 3.2 is a necessary and sufficient condition for to have the Grundy number , and we can use all the lemmas and theorems in previous sections for the function and .

Definition 4.1. We define the function as the followings.
Let be a natural number such that

Let for any .

We are going to show that the condition in Definition 4.1 is a necessary and sufficient condition for the chocolate bar to have the Grundy number .

Lemma 4.1. Let and be a natural number. Then

 (4.5)

if and only if there exist a natural number and a non-negative integer such that

 (4.6)

Proof. We suppose Relation (4.5). When , let with and . Let with and . Since and , . Since , . Therefore, for . Let and , then we have Relation (4.6). When , we let and . Then we have Relation (4.6).

Next we suppose that there exist a natural number and a non-negative integer that satisfy Relation (4.6). Then it is clear that we have Relation (4.5).

Lemma 4.2. Let and be a natural number such that for . Let such that . Then

 (4.7)

Proof. By Lemma 4.1, there exist a natural number and a non-negative integer such that and .

Let such that . Then we write in base 2, and we have . We prove that for . Let such that

. Then there exist such that and for . Therefore,

 (4.8)

Since for such that , the inequality in (4.8) implies Relation (4.7).

Lemma 4.3. Let be a natural number, and for . Suppose that for . Then if and only if .

Proof. Suppose that . By Lemma 1.1, , and hence . Let such that . Then , and Lemma 1.1 implies . Clearly for some natural number , and hence . By Lemma 1.1, we have .
Conversely we suppose that . Then, Lemma 1.1 implies for any , and hence for any . For any such that , Lemma 1.1 implies that there exists such that and . By Lemma 1.1, there exists such that , and hence we have . Therefore Lemma 1.1 implies .

Lemma 4.4. Let be a natural number such that

 (4.9)

Then, for any such that , we have

 (4.10)

In particular .

Proof. By Theorem 3.2 and the definition of Grundy number,

 (4.11)

When , we have , and hence . Since and , Lemma 3.10, Equation (4.9) and Lemma 4.2 imply
. Hence, by Theorem 3.2,

 (4.12)

Equation (4.11) and Equation (4.12) imply Equation (4.10). Therefore, by Lemma 1, we have .

Lemma 4.5. Let be a natural number such that

 (4.13)

For any such that , we have

 (4.14)

Proof. By Relation (4.13) and Lemma 4.4, we have

 (4.15)

for any such that .

Since , Lemma 4.4 implies that for

 (4.16)

Since , Lemma 4.4 implies that for

 (4.17)

Lemma 4.3, the inequality in (4.16), the inequality in (4.17) and Equation (4.15) imply (4.14). We have completed the proof.

Theorem 4.1. Let be a natural number such that

 (4.18)

and for any . Let be the Grundy number of . Then for any such that . Let such that . We prove by mathematical induction, and we assume that for such that or .

 (4.19)

By Lemma 4.5, , and hence we finish this proof.
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4.1 A Sufficient Condition for a Chocolate Bar to have the Grundy Number

Let be the function that satisfies the condition in Definition 3.2 and let be the Grundy number of . The condition in Definition 3.2 is a necessary and sufficient condition for to have the Grundy number , and we can use all the lemmas and theorems in previous sections for the function and .

Definition 4.1. We define the function as the followings.
Let be a natural number such that

Let for any .

We are going to show that the condition in Definition 4.1 is a necessary and sufficient condition for the chocolate bar to have the Grundy number .

Lemma 4.1. Let and be a natural number. Then

 (4.5)

if and only if there exist a natural number and a non-negative integer such that

 (4.6)

Proof. We suppose Relation (4.5). When , let with and . Let with and . Since and , . Since , . Therefore, for . Let and , then we have Relation (4.6). When , we let and . Then we have Relation (4.6).

Next we suppose that there exist a natural number and a non-negative integer that satisfy Relation (4.6). Then it is clear that we have Relation (4.5).

Lemma 4.2. Let and be a natural number such that for . Let such that . Then

 (4.7)

Proof. By Lemma 4.1, there exist a natural number and a non-negative integer such that and .

Let such that . Then we write in base 2, and we have . We prove that for . Let such that

. Then there exist such that and for . Therefore,

 (4.8)

Since for such that , the inequality in (4.8) implies Relation (4.7).

Lemma 4.3. Let be a natural number, and for . Suppose that for . Then if and only if .

Proof. Suppose that . By Lemma 1.1, , and hence . Let such that . Then , and Lemma 1.1 implies . Clearly for some natural number , and hence . By Lemma 1.1, we have .
Conversely we suppose that . Then, Lemma 1.1 implies for any , and hence for any . For any such that , Lemma 1.1 implies that there exists such that and . By Lemma 1.1, there exists such that , and hence we have . Therefore Lemma 1.1 implies .

Lemma 4.4. Let be a natural number such that

 (4.9)

Then, for any such that , we have

 (4.10)

In particular .

Proof. By Theorem 3.2 and the definition of Grundy number,

 (4.11)

When , we have , and hence . Since and , Lemma 3.10, Equation (4.9) and Lemma 4.2 imply
. Hence, by Theorem 3.2,

 (4.12)

Equation (4.11) and Equation (4.12) imply Equation (4.10). Therefore, by Lemma 1, we have .

Lemma 4.5. Let be a natural number such that

 (4.13)

For any such that , we have

 (4.14)

Proof. By Relation (4.13) and Lemma 4.4, we have

 (4.15)

for any such that .

Since , Lemma 4.4 implies that for

 (4.16)

Since , Lemma 4.4 implies that for

 (4.17)

Lemma 4.3, the inequality in (4.16), the inequality in (4.17) and Equation (4.15) imply (4.14). We have completed the proof.

Theorem 4.1. Let be a natural number such that

 (4.18)

and for any . Let be the Grundy number of . Then for any such that . Let such that . We prove by mathematical induction, and we assume that for such that or .

 (4.19)

By Lemma 4.5, , and hence we finish this proof.
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The text passage is a detailed mathematical proof involving the Grundy number of a chocolate bar, with several lemmas and a theorem. It does not contain any explicit mathematical questions or answers to such questions.

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