金声玉亮2.0模型评测报告

问题

You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format "<expression Y> = $<latex code>$" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>.

The question is:
Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$.
-Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$.

The solution is:
We claim the answer is $(\frac{n+1}{2})^2$ .
-We claim the answer is $(\frac{n+1}{2})^2$ .

First, consider a checkerboard tiling of the board with 4 colors: R, G, B, Y. Number each column from $1$ to $n$ from left to right and each row from $1$ to $n$ from top to bottom. We color a tile R if its row and column are odd, a tile G is its row is even but its column is odd, a tile B if its row and column is even, and a tile Y if its row is odd but its column is even.

Lemma 1: Throughout our moves, the color of the uncolored tile stays an invariant.

Consider that a domino can either only change rows or can only change columns. Therefore, sliding a domino into the hole and creating a new one has two possible colors. Of these, note that the new hole will always trivially be two tiles away from the old hole, meaning that the parity of both the row and column number stays the same. Thus, the lemma holds.

Lemma 2: There are more red tiles than any other color. Because each color is uniquely defined by the parity of a pair of column and row number, it satisfies to show that given an odd integer $n$ , there are more odd positive integers less than or equal to $n$ than even ones. Obviously, this is true, and so red will have more tiles than any other color.

Lemma 3: For any starting configuration $C$ and any blank tile $B$ such that the blank tile's color matches the blank tile's color of $C$ , there is no more than one unique configuration $C'$ that can be produced from $C$ using valid moves.

We will use proof by contradiction. Assume there exists two different $C'$ . We can get from one of these $C'$ to another using moves. However, we have to finish off with the same hole as before. Before the last move, the hole must be two tiles away from the starting hole. However, because the domino we used to move into the blank tile's spot is in the way, that hole must be congruent to the hole produced after the first move. We can induct this logic, and because there is a limited amount of tiles with the same color, eventually we will run out of tiles to apply this to. Therefore, having two distinct $C'$ with the same starting hole $B$ is impossible with some $C$ .

We will now prove that $(\frac{n+1}{2})^2$ is the answer. There are <missing 1> rows and $\frac{n+1}{2}$ columns that are odd, and thus there are $(\frac{n+1}{2})^2$ red tiles. Given lemma 3, this is our upper bound for a maximum. To establish that $(\frac{n+1}{2})^2$ is indeed possible, we construct such a $C$ :

In the first column, leave the first tile up blank. Then, continuously fill in vertically oriented dominos in that column until it reaches the bottom. In the next $n-1$ columns, place $\frac{n-1}{2}$ vertically oriented dominos in a row starting from the top. At the bottom row, starting with the first unfilled tile on the left, place horizontally aligned dominos in a row until you reach the right.

Obviously, the top left tile is red. It suffices to show that any red tile may be uncovered. For the first column, one may slide some dominos on the first column until the desired tile is uncovered. For the bottom row, all the first dominos may be slid up, and then the bottom dominos may be slid to the left until the desired red tile is uncovered. Finally, for the rest of the red tiles, the bottom red tile in the same color may be revealed, and then vertically aligned dominos in the same column may be slid down until the desired tile is revealed. Therefore, this configuration may produce $(\frac{n+1}{2})^2$ different configurations with moves.

Hence, we have proved that $(\frac{n+1}{2})^2$ is the maximum, and we are done.

The formulae are:
<expression 1> \frac{n+1}{2}

Your final answer should be STRICTLY in the format:

<Detailed reasoning>

Answer: <comma separated list of numbers representing expression identifiers>

答案

模型	模型答案	正确答案
金声玉亮2.0	<Detailed reasoning> To solve this problem, we need to match the formulae to the missing tags in the solution. The solution has only one missing tag, which is <missing 1>. This tag is related to the number of rows that are odd. Looking at the formulae, we have <expression 1> = \frac{n+1}{2}, which represents the number of odd rows or columns. Therefore, the missing tag <missing 1> should be filled with expression 1. Answer: 1	1
deepseek-r1	<Detailed reasoning> The missing part in the solution states that there are "<missing 1> rows and (n+1)/2 columns that are odd". Since the number of odd rows in an odd n x n grid is (n+1)/2 (as there are (n+1)/2 odd-numbered rows when n is odd), the missing formula here is (n+1)/2. This corresponds to expression 1 in the provided formulae. Hence, <missing 1> should be filled with expression 1. </Answer> Answer: 1	1
gpto1	To determine which expression fills in for `<missing 1>`, let's examine the sentence where it appears: "We will now prove that \((\frac{n+1}{2})^2\) is the answer. There are `<missing 1>` rows and \(\frac{n+1}{2}\) columns that are odd, and thus there are \((\frac{n+1}{2})^2\) red tiles." Since \(n\) is an odd positive integer, the number of odd rows from 1 to \(n\) is \(\frac{n+1}{2}\). Similarly, the number of odd columns is also \(\frac{n+1}{2}\). This matches the provided expression: - `<expression 1>` is \(\frac{n+1}{2}\) Therefore, `<missing 1>` should be filled with `<expression 1>`. Answer: 1	1
gpt-4o-2024-08-06	To solve this problem, we need to match the missing formulae in the solution with the given expressions. Let's go through the solution step by step and identify where the missing formulae should be placed. 1. Understanding the Problem: - We have an \( n \times n \) board with \( n \) being an odd positive integer. - A maximal grid-aligned configuration \( C \) consists of \((n^2-1)/2\) dominoes, covering all but one square. - We can slide dominoes to cover the uncovered square, creating new configurations. - We need to find the maximum number of distinct configurations \( k(C) \) obtainable from \( C \). 2. Solution Analysis: - The solution claims the answer is \((\frac{n+1}{2})^2\). - The solution uses a checkerboard coloring with four colors: R, G, B, Y. - Lemma 1 states that the color of the uncovered tile remains invariant. - Lemma 2 states there are more red tiles than any other color. - Lemma 3 states that for any starting configuration \( C \) and any blank tile \( B \) of the same color, there is at most one unique configuration \( C' \). 3. Identifying Missing Formulae: - The missing formula in the solution is in the statement: "There are <missing 1> rows and \(\frac{n+1}{2}\) columns that are odd, and thus there are \((\frac{n+1}{2})^2\) red tiles." 4. Matching with Given Expressions: - We have one expression given: - <expression 1> = \(\frac{n+1}{2}\) 5. Filling the Missing Formula: - The missing part "<missing 1>" should represent the number of odd rows, which is the same as the number of odd columns since \( n \) is odd. Therefore, it should be \(\frac{n+1}{2}\), which matches <expression 1>. 6. Conclusion: - The missing formula <missing 1> should be filled with <expression 1>. Detailed reasoning: The solution involves counting the number of red tiles on the board, which are determined by the number of odd rows and columns. Since \( n \) is odd, the number of odd rows and columns is \(\frac{n+1}{2}\). This matches the given expression <expression 1>. Therefore, the missing part in the solution should be filled with this expression. Answer: 1	1
DeepSeek-Coder-V2-Lite-Instruct	<Detailed reasoning> Answer: 1, 2, 3	1
Meta-Llama-3.1-405B-Instruct-Turbo	To solve this problem, we need to match the formulae to the missing tags in the solution. The solution has only one missing tag, which is <missing 1>. This tag is related to the number of rows that are odd. Looking at the formulae, we have <expression 1> = \frac{n+1}{2}, which represents the number of odd rows or columns. Therefore, the missing tag <missing 1> should be filled with expression 1. Answer: 1	1
Meta-Llama-3.1-70B-Instruct-Turbo	To solve this problem, we need to match the formulae to the missing tags in the solution. The first missing tag is <missing 1>, which is used to describe the number of rows that are odd. Looking at the formulae, we see that <expression 1> is \frac{n+1}{2}, which represents the number of odd rows (or columns) in the grid. Therefore, <missing 1> should be filled with <expression 1>. There are no other missing tags in the solution, so our answer is simply the identifier of the first expression. Answer: 1	1
Qwen2-72B-Instruct	The solution mentions that there are $(\frac{n+1}{2})^2$ rows and $\frac{n+1}{2}$ columns that are odd, which corresponds to the number of red tiles. This directly relates to the formula provided, which is <expression 1>. Therefore, <missing 1> should be filled with <expression 1>. Answer: 1	1
baichuan4	Detailed reasoning: 1. The solution mentions that there are <missing 1> rows and $\frac{n+1}{2}$ columns that are odd, and thus there are $(\frac{n+1}{2})^2$ red tiles. This suggests that <missing 1> should be the number of odd rows. Since $n$ is odd, there are $\frac{n+1}{2}$ odd rows. Therefore, <missing 1> should be filled by the formula that gives us $\frac{n+1}{2}$. 2. The only formula provided is <expression 1> = $\frac{n+1}{2}$. This matches the requirement for <missing 1>. Answer: 1	1
doubao	<Detailed reasoning>: In the text, it is mentioned that there are <missing 1> rows and $\frac{n+1}{2}$ columns that are odd, and it is later shown that there are $(\frac{n+1}{2})^2$ red tiles. So, <missing 1> should be filled with the expression for $\frac{n+1}{2}$. Answer: 1>	1
gpto1	To determine which expression fills in for `<missing 1>`, let's examine the sentence where it appears: "We will now prove that \((\frac{n+1}{2})^2\) is the answer. There are `<missing 1>` rows and \(\frac{n+1}{2}\) columns that are odd, and thus there are \((\frac{n+1}{2})^2\) red tiles." Since \(n\) is an odd positive integer, the number of odd rows from 1 to \(n\) is \(\frac{n+1}{2}\). Similarly, the number of odd columns is also \(\frac{n+1}{2}\). This matches the provided expression: - `<expression 1>` is \(\frac{n+1}{2}\) Therefore, `<missing 1>` should be filled with `<expression 1>`. Answer: 1	1
kimi	<Detailed reasoning> To fill in the missing parts of the solution, we need to identify the appropriate formula that matches the description given in the solution. The solution mentions that there are <missing 1> rows and $\frac{n+1}{2}$ columns that are odd, and thus there are $(\frac{n+1}{2})^2$ red tiles. The only formula provided is <expression 1> $\frac{n+1}{2}$. This formula is used to determine the number of odd rows and columns, which is exactly what is needed to calculate the number of red tiles. Therefore, the missing part should be filled with this formula. Answer: 1	1
minimax	To solve this problem, we need to match the missing formula in the solution with the given formulae. The missing formula is in the part of the solution that calculates the number of red tiles, which is also the upper bound for the maximum number of distinct configurations $k(C)$. The solution states that there are $\frac{n+1}{2}$ rows and $\frac{n+1}{2}$ columns that are odd, and thus there are $(\frac{n+1}{2})^2$ red tiles. This indicates that the missing formula should be the calculation of the number of red tiles, which is the square of the number of odd rows (or columns, since $n$ is odd). The given formulae are: <expression 1> \frac{n+1}{2} Since the missing formula should be the square of the number of odd rows (or columns), the correct formula to fill in the missing part is the one that represents the square of $\frac{n+1}{2}$. Therefore, the missing formula is: $$(\frac{n+1}{2})^2$$ This matches with <expression 1> because the expression identifier for the formula is 1. Answer: 1	1
wenxinyiyan	<Detailed reasoning> In the solution, there is only one missing tag <missing 1>. This tag appears in the context of discussing the number of odd rows and columns on the checkerboard tiling of the n-by-n board. The sentence reads: "There are <missing 1> rows and $\frac{n+1}{2}$ columns that are odd..." Since the board is n-by-n and n is an odd positive integer, it follows that there are $\frac{n+1}{2}$ odd-numbered rows and $\frac{n+1}{2}$ odd-numbered columns. The missing tag should be replaced by an expression representing the number of odd rows, which is the same as the number of odd columns, namely $\frac{n+1}{2}$. Comparing this to the provided formulae, we find that <expression 1> matches the needed expression, as it represents $\frac{n+1}{2}$. Answer: 1	1
zhipu	### Detailed Reasoning 1. Understanding the Solution Structure: - The solution involves proving that the maximum number of distinct maximal grid-aligned configurations, \( k(C) \), is \( \left(\frac{n+1}{2}\right)^2 \). - It uses several lemmas and a construction to establish the upper bound and show that this bound is achievable. 2. Identifying the Missing Formula: - The solution mentions that there are <missing 1> rows and \( \frac{n+1}{2} \) columns that are odd, and thus there are \( \left(\frac{n+1}{2}\right)^2 \) red tiles. - The context suggests that <missing 1> should be the number of odd rows, which is the same as the number of odd columns for an \( n \times n \) board. 3. Matching the Formulae: - Given the context, <missing 1> should represent \( \frac{n+1}{2} \), which is the number of odd rows (or columns) on the board. - The provided formulae include: - <expression 1> \( \frac{n+1}{2} \) 4. Conclusion: - The missing formula <missing 1> clearly matches <expression 1> since both refer to \( \frac{n+1}{2} \). ### Answer 1 This corresponds to the identifier of the expression that correctly fills the missing part in the solution.	1
deepseek-v3	<Detailed reasoning> The solution involves several steps and lemmas to establish that the maximum value of \( k(C) \) as a function of \( n \) is \( \left(\frac{n+1}{2}\right)^2 \). The key part where a formula is missing is in the sentence: "There are <missing 1> rows and \(\frac{n+1}{2}\) columns that are odd, and thus there are \(\left(\frac{n+1}{2}\right)^2\) red tiles." Here, the missing part refers to the number of rows that are odd. Since the board is \( n \)-by-\( n \) and \( n \) is odd, the number of odd rows is the same as the number of odd columns, which is \( \frac{n+1}{2} \). This matches the given formula: <expression 1> = \( \frac{n+1}{2} \). Therefore, the missing tag <missing 1> should be filled by expression 1. Answer: 1	1

模型

模型答案

正确答案

金声玉亮2.0

<Detailed reasoning>

To solve this problem, we need to match the formulae to the missing tags in the solution.

The solution has only one missing tag, which is <missing 1>. This tag is related to the number of rows that are odd.

Looking at the formulae, we have <expression 1> = \frac{n+1}{2}, which represents the number of odd rows or columns.

Therefore, the missing tag <missing 1> should be filled with expression 1.

Answer: 1

deepseek-r1

<Detailed reasoning>
The missing part in the solution states that there are "<missing 1> rows and (n+1)/2 columns that are odd". Since the number of odd rows in an odd n x n grid is (n+1)/2 (as there are (n+1)/2 odd-numbered rows when n is odd), the missing formula here is (n+1)/2. This corresponds to expression 1 in the provided formulae. Hence, <missing 1> should be filled with expression 1.
</Answer>

Answer: 1

gpto1

To determine which expression fills in for `<missing 1>`, let's examine the sentence where it appears:

"We will now prove that $(\frac{n+1}{2})^2$ is the answer. There are `<missing 1>` rows and $\frac{n+1}{2}$ columns that are odd, and thus there are $(\frac{n+1}{2})^2$ red tiles."

Since $n$ is an odd positive integer, the number of odd rows from 1 to $n$ is $\frac{n+1}{2}$. Similarly, the number of odd columns is also $\frac{n+1}{2}$. This matches the provided expression:

- `<expression 1>` is $\frac{n+1}{2}$

Therefore, `<missing 1>` should be filled with `<expression 1>`.

**Answer: 1**

gpt-4o-2024-08-06

To solve this problem, we need to match the missing formulae in the solution with the given expressions. Let's go through the solution step by step and identify where the missing formulae should be placed.

1. **Understanding the Problem:**
- We have an $ n \times n $ board with $ n $ being an odd positive integer.
- A maximal grid-aligned configuration $ C $ consists of $(n^2-1)/2$ dominoes, covering all but one square.
- We can slide dominoes to cover the uncovered square, creating new configurations.
- We need to find the maximum number of distinct configurations $ k(C) $ obtainable from $ C $.

2. **Solution Analysis:**
- The solution claims the answer is $(\frac{n+1}{2})^2$.
- The solution uses a checkerboard coloring with four colors: R, G, B, Y.
- Lemma 1 states that the color of the uncovered tile remains invariant.
- Lemma 2 states there are more red tiles than any other color.
- Lemma 3 states that for any starting configuration $ C $ and any blank tile $ B $ of the same color, there is at most one unique configuration $ C' $.

3. **Identifying Missing Formulae:**
- The missing formula in the solution is in the statement: "There are <missing 1> rows and $\frac{n+1}{2}$ columns that are odd, and thus there are $(\frac{n+1}{2})^2$ red tiles."

4. **Matching with Given Expressions:**
- We have one expression given:
- <expression 1> = $\frac{n+1}{2}$

5. **Filling the Missing Formula:**
- The missing part "<missing 1>" should represent the number of odd rows, which is the same as the number of odd columns since $ n $ is odd. Therefore, it should be $\frac{n+1}{2}$, which matches <expression 1>.

6. **Conclusion:**
- The missing formula <missing 1> should be filled with <expression 1>.

**Detailed reasoning:**
The solution involves counting the number of red tiles on the board, which are determined by the number of odd rows and columns. Since $ n $ is odd, the number of odd rows and columns is $\frac{n+1}{2}$. This matches the given expression <expression 1>. Therefore, the missing part in the solution should be filled with this expression.

**Answer:** 1

DeepSeek-Coder-V2-Lite-Instruct

<Detailed reasoning>

Answer: 1, 2, 3

Meta-Llama-3.1-405B-Instruct-Turbo

To solve this problem, we need to match the formulae to the missing tags in the solution.

The solution has only one missing tag, which is <missing 1>. This tag is related to the number of rows that are odd.

Looking at the formulae, we have <expression 1> = \frac{n+1}{2}, which represents the number of odd rows or columns.

Therefore, the missing tag <missing 1> should be filled with expression 1.

Answer: 1

Meta-Llama-3.1-70B-Instruct-Turbo

To solve this problem, we need to match the formulae to the missing tags in the solution.

The first missing tag is <missing 1>, which is used to describe the number of rows that are odd. Looking at the formulae, we see that <expression 1> is \frac{n+1}{2}, which represents the number of odd rows (or columns) in the grid. Therefore, <missing 1> should be filled with <expression 1>.

There are no other missing tags in the solution, so our answer is simply the identifier of the first expression.

Answer: 1

Qwen2-72B-Instruct

The solution mentions that there are $(\frac{n+1}{2})^2$ rows and $\frac{n+1}{2}$ columns that are odd, which corresponds to the number of red tiles. This directly relates to the formula provided, which is <expression 1>. Therefore, <missing 1> should be filled with <expression 1>.

Answer: 1

baichuan4

Detailed reasoning:

1. The solution mentions that there are <missing 1> rows and $\frac{n+1}{2}$ columns that are odd, and thus there are $(\frac{n+1}{2})^2$ red tiles. This suggests that <missing 1> should be the number of odd rows. Since $n$ is odd, there are $\frac{n+1}{2}$ odd rows. Therefore, <missing 1> should be filled by the formula that gives us $\frac{n+1}{2}$.

2. The only formula provided is <expression 1> = $\frac{n+1}{2}$. This matches the requirement for <missing 1>.

Answer: 1

doubao

<Detailed reasoning>: In the text, it is mentioned that there are <missing 1> rows and $\frac{n+1}{2}$ columns that are odd, and it is later shown that there are $(\frac{n+1}{2})^2$ red tiles. So, <missing 1> should be filled with the expression for $\frac{n+1}{2}$.
Answer: 1>

gpto1

kimi

<Detailed reasoning>

To fill in the missing parts of the solution, we need to identify the appropriate formula that matches the description given in the solution. The solution mentions that there are <missing 1> rows and $\frac{n+1}{2}$ columns that are odd, and thus there are $(\frac{n+1}{2})^2$ red tiles.

The only formula provided is <expression 1> $\frac{n+1}{2}$. This formula is used to determine the number of odd rows and columns, which is exactly what is needed to calculate the number of red tiles.

Therefore, the missing part should be filled with this formula.

Answer: 1

minimax

To solve this problem, we need to match the missing formula in the solution with the given formulae. The missing formula is in the part of the solution that calculates the number of red tiles, which is also the upper bound for the maximum number of distinct configurations $k(C)$.

The solution states that there are $\frac{n+1}{2}$ rows and $\frac{n+1}{2}$ columns that are odd, and thus there are $(\frac{n+1}{2})^2$ red tiles. This indicates that the missing formula should be the calculation of the number of red tiles, which is the square of the number of odd rows (or columns, since $n$ is odd).

The given formulae are:
<expression 1> \frac{n+1}{2}

Since the missing formula should be the square of the number of odd rows (or columns), the correct formula to fill in the missing part is the one that represents the square of $\frac{n+1}{2}$.

Therefore, the missing formula is:
$$(\frac{n+1}{2})^2$$

This matches with <expression 1> because the expression identifier for the formula is 1.

Answer: 1

wenxinyiyan

<Detailed reasoning>
In the solution, there is only one missing tag <missing 1>. This tag appears in the context of discussing the number of odd rows and columns on the checkerboard tiling of the n-by-n board. The sentence reads: "There are <missing 1> rows and $\frac{n+1}{2}$ columns that are odd..." Since the board is n-by-n and n is an odd positive integer, it follows that there are $\frac{n+1}{2}$ odd-numbered rows and $\frac{n+1}{2}$ odd-numbered columns. The missing tag should be replaced by an expression representing the number of odd rows, which is the same as the number of odd columns, namely $\frac{n+1}{2}$.

Comparing this to the provided formulae, we find that <expression 1> matches the needed expression, as it represents $\frac{n+1}{2}$.

Answer: 1

zhipu

### Detailed Reasoning

1. **Understanding the Solution Structure**:
- The solution involves proving that the maximum number of distinct maximal grid-aligned configurations, $ k(C) $, is $ \left(\frac{n+1}{2}\right)^2 $.
- It uses several lemmas and a construction to establish the upper bound and show that this bound is achievable.

2. **Identifying the Missing Formula**:
- The solution mentions that there are <missing 1> rows and $ \frac{n+1}{2} $ columns that are odd, and thus there are $ \left(\frac{n+1}{2}\right)^2 $ red tiles.
- The context suggests that <missing 1> should be the number of odd rows, which is the same as the number of odd columns for an $ n \times n $ board.

3. **Matching the Formulae**:
- Given the context, <missing 1> should represent $ \frac{n+1}{2} $, which is the number of odd rows (or columns) on the board.
- The provided formulae include:
- <expression 1> $ \frac{n+1}{2} $

4. **Conclusion**:
- The missing formula <missing 1> clearly matches <expression 1> since both refer to $ \frac{n+1}{2} $.

### Answer

1

This corresponds to the identifier of the expression that correctly fills the missing part in the solution.

deepseek-v3

<Detailed reasoning>

The solution involves several steps and lemmas to establish that the maximum value of $ k(C) $ as a function of $ n $ is $ \left(\frac{n+1}{2}\right)^2 $. The key part where a formula is missing is in the sentence:

"There are <missing 1> rows and $\frac{n+1}{2}$ columns that are odd, and thus there are $\left(\frac{n+1}{2}\right)^2$ red tiles."

Here, the missing part refers to the number of rows that are odd. Since the board is $ n $-by-$ n $ and $ n $ is odd, the number of odd rows is the same as the number of odd columns, which is $ \frac{n+1}{2} $. This matches the given formula:

<expression 1> = $ \frac{n+1}{2} $.

Therefore, the missing tag <missing 1> should be filled by expression 1.

Answer: 1