**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Action Figures
time_limit_ms: 2500
memory_limit_mb: 512
Description: There is a shop that sells action figures near Monocarp's house. A new set of action figures will be released shortly; this set contains $n$ figures, the $i$-th figure costs $i$ coins and is available for purchase from day $i$ to day $n$.

For each of the $n$ days, Monocarp knows whether he can visit the shop.

Every time Monocarp visits the shop, he can buy any number of action figures which are sold in the shop (of course, he cannot buy an action figure that is not yet available for purchase). If Monocarp buys at least two figures during the same day, he gets a discount equal to the cost of the most expensive figure he buys (in other words, he gets the most expensive of the figures he buys for free).

Monocarp wants to buy exactly one $1$-st figure, one $2$-nd figure, ..., one $n$-th figure from the set. He cannot buy the same figure twice. What is the minimum amount of money he has to spend?

Input: The first line contains one integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

Each test case consists of two lines:

  * the first line contains one integer $n$ ($1 \le n \le 4 \cdot 10^5$) — the number of figures in the set (and the number of days);    * the second line contains a string $s$ ($|s| = n$, each $s_i$ is either 0 or 1). If Monocarp can visit the shop on the $i$-th day, then $s_i$ is 1; otherwise, $s_i$ is 0.

Additional constraints on the input:

  * in each test case, $s_n$ is 1, so Monocarp is always able to buy all figures during the $n$-th day;    * the sum of $n$ over all test cases does not exceed $4 \cdot 10^5$.

Output: For each test case, print one integer — the minimum amount of money Monocarp has to spend.

Examples:
input:
4
1
1
6
101101
7
1110001
5
11111
output:
1
8
18
6

Note: In the first test case, Monocarp buys the $1$-st figure on the $1$-st day and spends $1$ coin.

In the second test case, Monocarp can buy the $1$-st and the $3$-rd figure on the $3$-rd day, the $2$-nd and the $4$-th figure on the $4$-th day, and the $5$-th and the $6$-th figure on the $6$-th day. Then, he will spend $1+2+5=8$ coins.

In the third test case, Monocarp can buy the $2$-nd and the $3$-rd figure on the $3$-rd day, and all other figures on the $7$-th day. Then, he will spend $1+2+4+5+6 = 18$ coins.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;

void populateVectors(const string& s, vector<int>& prevzeros, vector<int>& prevones) {
    int n = s.size();
    for (int i = 0; i < n - 1; i++) {
        if (s[i] == '0')
            prevzeros.push_back(i);
        else
            prevones.push_back(i);
    }
    reverse(prevones.begin(), prevones.end());
}

int calculateTotal(int n) {
    return n * (n + 1) / 2;
}

int adjustTotal(const string& s, vector<int>& prevzeros, vector<int>& prevones, int total) {
    int pzi = prevzeros.size() - 1;
    int poi = prevones.size() - 1;
    for (int i = s.size() - 1; i >= 0; i--) {
        while (pzi >= 0 && prevzeros[pzi] >= i)
            pzi--;
        while (poi >= 0 && prevones[poi] >= i)
            poi--;

        if (s[i] == '1') {
            if (pzi >= 0) {
                total -= (i + 1);
                pzi--;
            } else if (poi >= 0) {
                total -= (i + 1);
                poi--;
            }
        }
    }
    return total;
}

void solve() {
    int n;
    cin >> n;
    string s;
    cin >> s;

    vector<int> prevzeros;
    vector<int> prevones;

    populateVectors(s, prevzeros, prevones);
    int total = calculateTotal(n);
    total = adjustTotal(s, prevzeros, prevones, total);

    cout << total << endl;
}

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    int t;
    cin >> t;
    while (t--) {
        solve();
    }
    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to consider edge cases and scenarios that can highlight the issues in the logic. The code's main logic involves calculating the total cost and then adjusting it based on the days Monocarp can visit the shop. The bug might be related to how the code handles the discounts and the order of purchases.

### Key Points to Consider:
1. **Single Visit Days**: Days where Monocarp can only visit once.
2. **Multiple Visit Days**: Days where Monocarp can visit multiple times.
3. **Edge Cases**: Small values of \( n \) and specific patterns in the string \( s \).

### Challenging Test Input:
Let's create a test input where the string \( s \) has a pattern that forces the code to handle discounts incorrectly. Specifically, we can create a string where Monocarp can visit the shop on days that are not optimal for the discount logic.

```plaintext
4
3
101
4
1011
5
10111
6
101111
```

### Explanation:
1. **Test Case 1: \( n = 3 \), \( s = "101" \)**
   - Monocarp can visit on days 1 and 3.
   - He can buy the 1st figure on day 1 and the 2nd and 3rd figures on day 3.
   - Total cost: \( 1 + 2 = 3 \) (the 3rd figure is free).

2. **Test Case 2: \( n = 4 \), \( s = "1011" \)**
   - Monocarp can visit on days 1, 3, and 4.
   - He can buy the 1st figure on day 1, the 2nd and 3rd figures on day 3, and the 4th figure on day 4.
   - Total cost: \( 1 + 2 + 4 = 7 \) (the 3rd figure is free).

3. **Test Case 3: \( n = 5 \), \( s = "10111" \)**
   - Monocarp can visit on days 1, 3, 4, and 5.
   - He can buy the 1st figure on day 1, the 2nd and 3rd figures on day 3, and the 4th and 5th figures on day 5.
   - Total cost: \( 1 + 2 + 4 = 7 \) (the 3rd and 5th figures are free).

4. **Test Case 4: \( n = 6 \), \( s = "101111" \)**
   - Monocarp can visit on days 1, 3, 4, 5, and 6.
   - He can buy the 1st figure on day 1, the 2nd and 3rd figures on day 3, and the 4th, 5th, and 6th figures on day 6.
   - Total cost: \( 1 + 2 + 4 = 7 \) (the 3rd and 6th figures are free).

These test cases are designed to highlight the issues in the discount logic and the order of purchases. The small scale ensures that the input is manageable and complete.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Action Figures
time_limit_ms: 2500
memory_limit_mb: 512
Description: There is a shop that sells action figures near Monocarp's house. A new set of action figures will be released shortly; this set contains $n$ figures, the $i$-th figure costs $i$ coins and is available for purchase from day $i$ to day $n$.

For each of the $n$ days, Monocarp knows whether he can visit the shop.

Every time Monocarp visits the shop, he can buy any number of action figures which are sold in the shop (of course, he cannot buy an action figure that is not yet available for purchase). If Monocarp buys at least two figures during the same day, he gets a discount equal to the cost of the most expensive figure he buys (in other words, he gets the most expensive of the figures he buys for free).

Monocarp wants to buy exactly one $1$-st figure, one $2$-nd figure, ..., one $n$-th figure from the set. He cannot buy the same figure twice. What is the minimum amount of money he has to spend?

Input: The first line contains one integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

Each test case consists of two lines:

  * the first line contains one integer $n$ ($1 \le n \le 4 \cdot 10^5$) — the number of figures in the set (and the number of days);    * the second line contains a string $s$ ($|s| = n$, each $s_i$ is either 0 or 1). If Monocarp can visit the shop on the $i$-th day, then $s_i$ is 1; otherwise, $s_i$ is 0.

Additional constraints on the input:

  * in each test case, $s_n$ is 1, so Monocarp is always able to buy all figures during the $n$-th day;    * the sum of $n$ over all test cases does not exceed $4 \cdot 10^5$.

Output: For each test case, print one integer — the minimum amount of money Monocarp has to spend.

Examples:
input:
4
1
1
6
101101
7
1110001
5
11111
output:
1
8
18
6

Note: In the first test case, Monocarp buys the $1$-st figure on the $1$-st day and spends $1$ coin.

In the second test case, Monocarp can buy the $1$-st and the $3$-rd figure on the $3$-rd day, the $2$-nd and the $4$-th figure on the $4$-th day, and the $5$-th and the $6$-th figure on the $6$-th day. Then, he will spend $1+2+5=8$ coins.

In the third test case, Monocarp can buy the $2$-nd and the $3$-rd figure on the $3$-rd day, and all other figures on the $7$-th day. Then, he will spend $1+2+4+5+6 = 18$ coins.

Buggy Code:
import java.util.*;

public class Main {
  static boolean LOCAL = System.getProperty("ONLINE_JUDGE")==null;
    static void dbg(Object... o){if(LOCAL)System.err.println(Arrays.deepToString(o));}
    static long maxi;
    public static void main(String[] args) {
      Scanner sc = new Scanner(System.in);
       int t = sc.nextInt();
       while(t --> 0)
       {
         int n = sc.nextInt();
         int conut = 0;
         String s = sc.next();
         int sum = n * (n + 1) / 2;

         for(int i = n - 1;i >= 0;i--)
         {
           if(conut >= i) break;

          if(s.charAt(i) == '1')
           {
            conut++;
            sum = sum -  (i + 1);
           }
          if(s.charAt(i) == '0')
          {
            conut--;
            if(conut < 0) conut = 0;
          }

         }

         System.out.println(sum);

         //We have max right

      }
    }

}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3
4
1111
5
10101
6
101101
```

### Explanation:

1. **Test Case 1:**
   - **Input:** `4 1111`
   - **Expected Output:** `10`
   - **Reason:** Monocarp can buy all figures on the 4th day. The total cost without any discount is `1 + 2 + 3 + 4 = 10`. Since he can buy all figures on the same day, he gets the most expensive figure (4) for free, so the total cost is `10 - 4 = 6`.

2. **Test Case 2:**
   - **Input:** `5 10101`
   - **Expected Output:** `11`
   - **Reason:** Monocarp can buy the 1st and 3rd figures on the 3rd day, the 2nd and 4th figures on the 4th day, and the 5th figure on the 5th day. The total cost without any discount is `1 + 2 + 3 + 4 + 5 = 15`. The discounts are 3 (from the 3rd day) and 4 (from the 4th day), so the total cost is `15 - 3 - 4 = 8`.

3. **Test Case 3:**
   - **Input:** `6 101101`
   - **Expected Output:** `8`
   - **Reason:** Monocarp can buy the 1st and 3rd figures on the 3rd day, the 2nd and 4th figures on the 4th day, and the 5th and 6th figures on the 6th day. The total cost without any discount is `1 + 2 + 3 + 4 + 5 + 6 = 21`. The discounts are 3 (from the 3rd day), 4 (from the 4th day), and 6 (from the 6th day), so the total cost is `21 - 3 - 4 - 6 = 8`.

These test cases are designed to expose the bug in the given code by ensuring that the logic for counting the number of days Monocarp can visit the shop and applying the discount is correctly handled. The buggy code incorrectly decrements the count of available days even when Monocarp cannot visit the shop, which can lead to incorrect results.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
#include<cstdio>

#define N 30

int a[N][N];

void solve()
{
	int n;
	scanf("%d",&n);
	int i,j;

	int k=0;
	for(i=0;i<n*2;i+=2)
	{
		int flag = 0;
		if(!k)
		{
			for(j=0;j<n*2;j+=2)
				if(!flag)
				{
					a[i][j] =a[i][j+1] =a[i+1][j]=a[i+1][j+1]='#';
					flag = 1;

				}
				else
				{
					a[i][j] =a[i][j+1] =a[i+1][j]=a[i+1][j+1]='.';
					flag = 0;
				}
				k=1;
		}
		else
		{
			for(j=0;j<n*2;j+=2)
				if(!flag)
				{
					a[i][j] =a[i][j+1] =a[i+1][j]=a[i+1][j+1]='.';
					flag = 1;

				}
				else
				{
					a[i][j] =a[i][j+1] =a[i+1][j]=a[i+1][j+1]='#';
					flag = 0;
				}
				k=0;
		}
	}

	for(i=0;i<n*2;i++)
	{
		for(j=0;j<n*2;j++)
			printf("%c",a[i][j]);

		puts("");
	}

}

int main()
{
	int T;

	scanf("%d",&T);

	while(T--) solve();

	return 0;

}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to focus on edge cases and scenarios that might cause the code to fail. The key points to consider are:

1. **Single Test Case with Minimum Input**: This will help us understand if the basic functionality is working.
2. **Single Test Case with Maximum Input**: This will test the limits of the code and ensure it handles large inputs correctly.
3. **Multiple Test Cases with Varying Inputs**: This will test the code's ability to handle different sizes of checkerboards and ensure consistency across multiple test cases.

Let's start with a small-scale test input that includes a mix of these scenarios:

```plaintext
3
1
2
3
```

### Explanation:
- **Test Case 1 (n=1)**: This is the smallest possible input. It will help us verify if the basic pattern is correct.
- **Test Case 2 (n=2)**: This is a slightly larger input. It will help us verify if the pattern is correctly repeated and if the alternating pattern works as expected.
- **Test Case 3 (n=3)**: This is a medium-sized input. It will help us verify if the code can handle a larger checkerboard and maintain the correct pattern.

This test input is small enough to be manageable but comprehensive enough to expose potential issues in the code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
#include <iostream>

using namespace std;

int n;
char Array[20][20];

void solve () {
	bool minidecision = 1;
	int cntr = 1;
	for (int i = 0; i < 2 * n; i +=2 ) {
		for (int j = 0; j < 2 * n; j += 2){
				if (minidecision) {
					Array[i][j] = '#';
					Array[i][j + 1] = '#';
					Array[i + 1][j] = '#';
					Array[i + 1][j + 1] = '#';
					minidecision = 0;
				}
				else {
					Array[i][j] = '.';
					Array[i][j + 1] = '.';
					Array[i + 1][j] = '.';
					Array[i + 1][j + 1] = '.';
					minidecision = 1;
				}

		}
		cntr++;
		if (cntr % 2 == 0) {
			minidecision = 0;
		}
		else {
			minidecision = 1;
		}
	}
	for (int i = 0; i < 2 * n; i++) {
		for (int j = 0; j < 2 * n; j++) {
			cout << Array[i][j];
		}
		cout << endl;
	}
}

int main() {
	int t; cin >> t;
	while (t--) {
		cin >> n;
		solve();
	}
	return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a test input that exposes the bug in the provided code, we need to focus on the logic that handles the checkerboard pattern. The bug in the code is that it does not correctly alternate the patterns between rows and columns. Specifically, the `minidecision` variable is used to decide whether to place `#` or `.` in a 2x2 block, but it is not correctly reset or toggled for each row and column.

### Steps to Generate the Test Input:
1. **Identify the Issue**: The `minidecision` variable is toggled in a way that does not ensure the correct checkerboard pattern.
2. **Edge Cases**: Small values of `n` (e.g., 1, 2, 3) are good to start with, as they can clearly show the pattern.
3. **Test Case Selection**: We should include at least one test case with a value of `n` that is likely to expose the issue, such as `n = 2`.

### Test Input:
```plaintext
1
2
```

### Explanation:
- **Test Case 1**: `n = 2`
  - The expected output for `n = 2` is:
    ```
    ##..
    ##..
    ..##
    ..##
    ```
  - The buggy code will output:
    ```
    ####
    ####
    ....
    ....
    ```
  - This clearly shows that the `minidecision` variable is not correctly alternating the patterns.

By using this test input, we can easily see that the checkerboard pattern is not being correctly generated, thus exposing the bug in the code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
#include<iostream>
#include<string>
#include<vector>

using namespace std;

char a[45][45] = {
	"##..##..##..##..##..##..##..##..##..##..",
	"##..##..##..##..##..##..##..##..##..##..",
	"..##..##..##..##..##..##..##..##..##..##",
	"..##..##..##..##..##..##..##..##..##..##",
	"##..##..##..##..##..##..##..##..##..##..",
	"##..##..##..##..##..##..##..##..##..##..",
	"..##..##..##..##..##..##..##..##..##..##",
	"..##..##..##..##..##..##..##..##..##..##",
	"##..##..##..##..##..##..##..##..##..##..",
	"##..##..##..##..##..##..##..##..##..##..",
	"..##..##..##..##..##..##..##..##..##..##",
	"..##..##..##..##..##..##..##..##..##..##",
	"##..##..##..##..##..##..##..##..##..##..",
	"##..##..##..##..##..##..##..##..##..##..",
	"..##..##..##..##..##..##..##..##..##..##",
	"..##..##..##..##..##..##..##..##..##..##",
	"##..##..##..##..##..##..##..##..##..##..",
	"##..##..##..##..##..##..##..##..##..##..",
	"..##..##..##..##..##..##..##..##..##..##",
	"..##..##..##..##..##..##..##..##..##..##",

};
void solve() {
	int n;
	cin >> n;
	for (int i = 0; i < 2 * n; i++) {
		for (int j = 0; j < 2 * n; j++) {
			cout << a[i][j];
		}
		cout << endl;
	}
}

int main() {
	ios::sync_with_stdio(0);
	cin.tie(0);
	int t;
	cin >> t;
	while (t--) {
		solve();
	}
	return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to consider the following:

1. **Edge Cases**: The smallest and largest possible values for \( n \) (i.e., 1 and 20).
2. **Pattern Misalignment**: The checkerboard pattern should alternate correctly, and any deviation from this pattern should be detectable.
3. **Multiple Test Cases**: Including multiple test cases to ensure the code handles different sizes of \( n \).

Given the constraints, a small-scale test input that includes both the smallest and a mid-range value for \( n \) would be effective. This will help us verify if the code correctly handles the smallest and a more complex case.

Here is the test input:

```plaintext
2
1
10
```

Explanation:
- The first test case with \( n = 1 \) is the smallest possible input, which should generate a 2x2 checkerboard.
- The second test case with \( n = 10 \) is a mid-range value, which should generate a 20x20 checkerboard.

This input will help us check if the code correctly handles both the smallest and a more complex checkerboard pattern.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
n=int(input())
for i in range(n):
    m=int(input())
    if m==1:
        print('##\n##')
    elif m==2:
        print('##..\n##..\n..##\n..##')
    elif m==3:
        print('##..##\n##..##\n..##..\n..##..\n##..##\n##..##')
    elif m==4:
        print('##..##..\n##..##..\n..##..##\n..##..##\n##..##..\n##..##..\n..##..##\n..##..##')

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on edge cases and scenarios that the code might handle incorrectly. The buggy code only handles specific values of `n` (1, 2, 3, and 4) explicitly. This means that any other value of `n` will not be handled correctly.

Given the constraints:
- \(1 \leq t \leq 20\) (number of test cases)
- \(1 \leq n \leq 20\) (value of `n` for each test case)

A good approach is to include values of `n` that are not explicitly handled by the buggy code, such as `5`, `6`, `7`, etc. Additionally, we can include the edge case of `n = 1` to ensure that the code handles the smallest possible input correctly.

Here is a small-scale test input that includes both edge cases and values not explicitly handled by the buggy code:

```plaintext
5
1
5
6
7
20
```

This test input includes:
- `n = 1` (smallest possible value, already handled correctly by the buggy code)
- `n = 5` (not explicitly handled, will expose the bug)
- `n = 6` (not explicitly handled, will expose the bug)
- `n = 7` (not explicitly handled, will expose the bug)
- `n = 20` (largest possible value, not explicitly handled, will expose the bug)

This should effectively highlight the limitations of the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
for _ in range(int(input())):
    N = int(input())
    if N == 1:
        print("##\n##")
    elif N == 2:
        print("##..\n##..\n..##\n..##")
    elif N == 3:
        print("##..##\n##..##\n..##..\n..##..\n##..##\n##..##")
    elif N == 4:
        print("##..##..\n##..##..\n..##..##\n..##..##\n##..##..\n##..##..\n..##..##\n..##..##")

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
4
1
2
3
4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
casos = int(input())
n = 0
tabla = [n]

while n != casos:
  n = n + 1
  tabla.append(n)
  tabla [n] = int(input())

n = 0
while n != casos:
  n = n + 1
  if tabla [n] == 1:
    print("##\n##")
  elif tabla [n] == 2:
    print("##..\n##..\n..##\n..##")
  elif tabla [n] == 3:
    print("##..##\n##..##\n..##..\n..##..\n##..##\n##..##")
  elif tabla [n] == 4:
    print("##..##..\n##..##..\n..##..##\n..##..##\n##..##..\n##..##..\n..##..##\n..##..##")
  else:
    pass

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, let's analyze the code and identify potential issues:

1. **Input Handling**: The code reads the number of test cases and then reads the value of `n` for each test case. However, the code incorrectly appends `n` to the `tabla` list and then uses it as an index, which can lead to issues if the input values are not sequential or if the list is not properly initialized.

2. **Checkerboard Generation**: The code uses a series of `if-elif-else` statements to handle specific values of `n`. This approach is inefficient and error-prone, especially for larger values of `n`.

3. **Edge Cases**: The code does not handle edge cases well, particularly for `n` values that are not explicitly checked (i.e., `n > 4`).

To expose these issues, we can create a test input that includes:
- A mix of small and large values of `n` to test the code's ability to handle different sizes of checkerboards.
- Repeated values of `n` to test the code's ability to handle multiple test cases with the same input.
- Values of `n` that are not explicitly checked (i.e., `n > 4`) to test the code's ability to handle unhandled cases.

Here is a small-scale test input that should expose the bug:

```plaintext
5
1
2
3
4
5
```

This test input includes:
- The smallest value of `n` (1) to check the basic functionality.
- Values 2, 3, and 4 to check the specific cases handled by the code.
- A value of `n` (5) that is not explicitly handled by the `if-elif-else` statements to test the code's ability to handle unhandled cases.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
import java.util.Scanner;

public class codeforce1950B
{
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);

        int t = sc.nextInt();
        for (int i = 1; i <=t; i++) {
            if (i == 1) {

                for (int j = 0; j < 2; j++) {
                    System.out.println("##");

                }
            }
            if (i == 2) {

                for (int j = 0; j < 2; j++) {
                    System.out.println("##..");
                }
                for (int j = 0; j < 2; j++) {
                    System.out.println("..##");
                }
            }

            if (i == 3) {

                for (int j = 0; j < 2; j++) {
                    System.out.println("##..##");
                }
                for (int j = 0; j < 2; j++) {
                    System.out.println("..##..");
                }
                for (int j = 0; j < 2; j++) {
                    System.out.println("##..##");
                }
            }

            if (i == 4) {

                for (int j1 = 0; j1 <2; j1++) {
                    System.out.println("##..##..");
                }

                for (int k = 0; k < 2; k++) {
                    System.out.println("..##..##");
                }

                for (int m = 0; m < 2; m++) {
                    System.out.println("##..##..");
                }

                for (int n = 0; n < 2; n++) {
                    System.out.println("..##..##");
                }

            }

        }
    }
    }

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on the following points:

1. **Edge Cases**: Test the smallest and largest valid values for \( n \) (i.e., \( n = 1 \) and \( n = 20 \)).
2. **Pattern Consistency**: Ensure the checkerboard pattern is consistent for different values of \( n \).
3. **General Case**: Test a few general values of \( n \) to ensure the pattern is correctly generated.

The buggy code provided has a specific implementation for \( n = 1, 2, 3, \) and \( 4 \). For other values of \( n \), it does not handle the general case correctly. Therefore, we should test values of \( n \) that are not explicitly handled in the code.

### Challenging Test Input

```plaintext
5
1
4
10
15
20
```

### Explanation

- **Test Case 1**: \( n = 1 \) (Edge Case)
  - This is the smallest valid value for \( n \).
- **Test Case 2**: \( n = 4 \)
  - This value is explicitly handled in the code, but it's good to verify the pattern.
- **Test Case 3**: \( n = 10 \)
  - This is a general value that is not explicitly handled in the code.
- **Test Case 4**: \( n = 15 \)
  - Another general value that is not explicitly handled in the code.
- **Test Case 5**: \( n = 20 \) (Edge Case)
  - This is the largest valid value for \( n \).

These test cases will help expose any issues in the code where the pattern generation for \( n \) is not correctly implemented for values other than 1, 2, 3, and 4.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
import java.util.*;
public class Main{
    public static void main(String[] args){
        Scanner s=new Scanner(System.in);
        int t=s.nextInt();
        while(t-->0){
            int n=s.nextInt();
            if(n==1) {
            	System.out.println("##");
            	System.out.println("##");
            }
            else {
            int c=n*2;
            for(int i=1;i<=n;i++) {

            	if(c>0) {
            	for(int j=1;j<=2;j++) {
            		c=c-1;
            		int temp1=n/2;
            		while(temp1>0) {
            			System.out.print("##");
            			System.out.print("..");
            			temp1--;
            			if(n%2==1) {
            				System.out.print("##");
            			}
            		}
            		System.out.println();

            	}
            	}
            	if(c>0) {
            	for(int k=1;k<=2;k++) {

            		int temp1=n/2;
            		c=c-1;
            		while(temp1>0) {
            			System.out.print("..");
            			System.out.print("##");
            			temp1--;
            			if(n%2==1) {
            				System.out.print("..");
            			}

            		}
            		System.out.println();

            	}
            	}
            }
            }
        }
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To expose the bug in the given buggy code, we need to focus on edge cases and scenarios that might cause the code to fail. The code seems to handle the basic cases correctly, but it might have issues with the pattern generation, especially when `n` is odd or when the pattern needs to be repeated in a specific way.

Let's analyze the code and identify potential issues:

1. **Pattern Generation for Odd and Even `n`**: The code has a specific way of handling the pattern for odd and even `n`. It uses `n/2` to determine the number of repetitions, which might cause issues when `n` is odd.
2. **Repetition and Pattern Consistency**: The code might not correctly handle the repetition of the `##..` and `..##` patterns, especially when `n` is large.

Given these points, a small-scale test input that can expose the bug would be:

- **Test Case 1**: `n = 1` (Basic case, should work correctly)
- **Test Case 2**: `n = 2` (Even `n`, should work correctly)
- **Test Case 3**: `n = 3` (Odd `n`, might expose the bug)

Let's create the test input:

```plaintext
3
1
2
3
```

This test input includes:
- A basic case (`n = 1`) to ensure the code works for the simplest scenario.
- An even `n` case (`n = 2`) to check the pattern for even numbers.
- An odd `n` case (`n = 3`) to check the pattern for odd numbers, which is more likely to expose the bug.

This should help identify any issues in the pattern generation for both even and odd values of `n`.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
#include <bits/stdc++.h>
using namespace std;

void generate_checkerboard(int n) {
    for (int i = 1; i <= 2 * n; ++i) {
        if (i == 1 || i == 2 || i == 5 || i == 6 || i == 9 || i == 10 || i == 13 || i == 14 || i == 17 || i == 18) {
            for (int j = 0; j < n; ++j) {
                if (j % 2 == 0)
                    cout << "##";
                else
                    cout << "..";
            }
            cout << endl;
        } else {
            for (int j = 0; j < n; ++j) {
                if (j % 2 == 0)
                    cout << "..";
                else
                    cout << "##";
            }
            cout << endl;
        }
    }
}

int main() {
    int t;
    cin >> t;
    while (t--) {
        int n;
        cin >> n;
        generate_checkerboard(n);
    }
    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the buggy code, we need to focus on the logic used to determine the pattern of the checkerboard. The buggy code has a specific condition that only handles certain rows (1, 2, 5, 6, 9, 10, 13, 14, 17, 18) differently. This condition is not general and will fail for larger values of \( n \).

### Step-by-Step Analysis:
1. **Identify the Bug**:
   - The code uses a hard-coded condition to handle specific rows (1, 2, 5, 6, 9, 10, 13, 14, 17, 18).
   - This condition is not scalable and will fail for rows that are not explicitly listed.

2. **Generate a Test Case**:
   - We need a test case where \( n \) is large enough to include rows that are not in the hard-coded list.
   - A small \( n \) that is not covered by the hard-coded list will be sufficient to expose the bug.

3. **Choose \( n \)**:
   - Let's choose \( n = 4 \). This will generate a \( 8 \times 8 \) checkerboard.
   - The rows 3, 4, 7, 8, 11, 12, 15, 16 are not in the hard-coded list and will use the else condition, which is incorrect for these rows.

### Test Input:
```plaintext
1
4
```

This test input will generate an \( 8 \times 8 \) checkerboard, and the rows that are not in the hard-coded list will expose the bug in the code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
#include <bits/stdc++.h>
#include <vector>
#include <string>
#include <set>
#include <map>
#include <numeric>
#define ll long long
#define f first
#define s second
using namespace std;
const ll maxN = int(1e6) + 5;
const ll BASE = int(1e9) +7;
ll n, t, d = 0, s = 0, sl, dl;
vector <ll> v;
char c[20][20];

void solve(){
	s = 0;
	d = 0;
	sl = 0;
	dl = 0;
	for (int i = 1;i <= 2 * n;++i){
		if (sl < 2){
			s = 0;
			d = 0;
			for (int j = 1;j <= 2 * n;++j){
				if (s < 2){
					c[i][j] = '#';
					s++;
				}
				else {
					c[i][j] = '.';
					d++;
				}
				if (s == 2 && d == 2){
					s = 0;
					d = 0;
				}
			}
			sl++;
		}
		else {
			s = 0;
			d = 0;
			for (int j = 1;j <=2 * n;++j){
				if (d < 2){
					c[i][j] = '.';
					d++;
				}
				else {
					c[i][j] = '#';
					s++;
				}
				if (s == 2 && d == 2){
					s = 0;
					d = 0;
				}
			}
			dl++;
		}
		if (sl == 2 && dl == 2){
			sl = 0;
			dl = 0;
		}
	}
	for (int i = 1;i <= 2 * n;++i){
		for (int j = 1;j <= 2 * n;++j){
			cout << c[i][j];
		}
		cout << "\n";
	}
}

void input(){
	cin >> t;
	while (t--){
		cin >> n;
		solve();
	}
}

int main(){
    ios_base::sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);
    input();
    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, let's analyze the code and identify potential issues:

### Analysis of the Buggy Code:
1. **Checkerboard Pattern Logic**:
   - The code attempts to create a checkerboard pattern by alternating between '#' and '.' in 2x2 blocks.
   - The logic for alternating the characters is flawed, especially when handling the 2x2 blocks.

2. **Potential Issues**:
   - The code uses `sl` and `dl` to keep track of the number of rows of '#' and '.', respectively. However, the reset logic for `sl` and `dl` is incorrect.
   - The code does not correctly handle the alternating pattern for larger values of `n`.

### Challenging Test Input:
To expose the bug, we need to test the code with values of `n` that are large enough to show the pattern issue but still small enough to fit within the output limits. We should also include edge cases and typical values.

### Test Input:
```plaintext
3
1
2
4
```

### Explanation:
1. **Test Case 1: \( n = 1 \)**
   - This is a simple case to ensure the basic pattern is correct.
   - Expected output:
     ```
     ##
     ##
     ```

2. **Test Case 2: \( n = 2 \)**
   - This is a slightly more complex case to check the 2x2 block pattern.
   - Expected output:
     ```
     ##..
     ##..
     ..##
     ..##
     ```

3. **Test Case 3: \( n = 4 \)**
   - This is a more challenging case to ensure the alternating pattern is correct for a larger board.
   - Expected output:
     ```
     ##..##..##
     ##..##..##
     ..##..##..
     ..##..##..
     ##..##..##
     ##..##..##
     ..##..##..
     ..##..##..
     ```

This test input should help in identifying the pattern issues in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
a=['#','#']
b=['.','.']
x=[a,a]
y=[b,b]
c=[[a,b],[a,b],[b,a],[b,a]]
d=[[a,b,a],[a,b,a],[b,a,b],[b,a,b],[a,b,a],[a,b,a]]
e=[[a,b,a,b],[a,b,a,b],[b,a,b,a],[b,a,b,a],[a,b,a,b],[a,b,a,b],[b,a,b,a],[b,a,b,a]]
count=0
ans=[x,c,d,e]
for i in ans:
    for j in i:
        for k in j:
            for l in k:
                print(l,end="")
        print()

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on the following aspects:

1. **Edge Cases**: Test with the smallest and largest possible values of `n` within the given constraints.
2. **Intermediate Values**: Test with values that are not trivial and might reveal issues in the pattern generation.
3. **Multiple Test Cases**: Ensure the test input contains multiple test cases to check the consistency of the solution.

Given the constraints:
- \(1 \leq t \leq 20\)
- \(1 \leq n \leq 20\)

Let's create a test input that includes:
- The smallest value of \(n\) (1)
- A value in the middle of the range (10)
- The largest value of \(n\) (20)
- A few intermediate values to ensure the pattern is correct

Here is the test input:

```plaintext
5
1
10
20
5
15
```

This test input includes:
- \(n = 1\): The smallest possible value.
- \(n = 10\): A middle value.
- \(n = 20\): The largest possible value.
- \(n = 5\): Another intermediate value.
- \(n = 15\): Another intermediate value.

These values will help expose any issues in the pattern generation and ensure the solution works correctly for a variety of input sizes.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
t = int(input())
for i in range(t):
    n=int(input())
    if n==1:
        print("##")
        print("##")
    elif n==2:
        print("##..")
        print("##..")
        print("..##")
        print("..##")
    elif n==3:
        print("##..##")
        print("##..##")
        print("..##..")
        print("..##..")
        print("##..##")
        print("##..##")
    elif n==4:
        print("##..##..")
        print("##..##..")
        print("..##..##")
        print("..##..##")
        print("##..##..")
        print("##..##..")
        print("..##..##")
        print("..##..##")
    elif n==5:
        print("##..##..##")
        print("##..##..##")
        print("..##..##..")
        print("..##..##..")
        print("##..##..##")
        print("##..##..##")
        print("..##..##..")
        print("..##..##..")
        print("##..##..##")
        print("##..##..##")
    elif n==6:
        print("##..##..##..")
        print("##..##..##..")
        print("..##..##..##")
        print("..##..##..##")
        print("##..##..##..")
        print("##..##..##..")
        print("..##..##..##")
        print("..##..##..##")
        print("##..##..##..")
        print("##..##..##..")
        print("..##..##..##")
        print("..##..##..##")
    elif n==7:
        print("##..##..##..##")
        print("##..##..##..##")
        print("..##..##..##..")
        print("..##..##..##..")
        print("##..##..##..##")
        print("##..##..##..##")
        print("..##..##..##..")
        print("..##..##..##..")
        print("##..##..##..##")
        print("##..##..##..##")
        print("..##..##..##..")
        print("..##..##..##..")
        print("##..##..##..##")
        print("##..##..##..##")
    elif n==8:
        print("##..##..##..##..")
        print("##..##..##..##..")
        print("..##..##..##..##")
        print("..##..##..##..##")
        print("##..##..##..##..")
        print("##..##..##..##..")
        print("..##..##..##..##")
        print("..##..##..##..##")
        print("##..##..##..##..")
        print("##..##..##..##..")
        print("..##..##..##..##")
        print("..##..##..##..##")
        print("##..##..##..##..")
        print("##..##..##..##..")
        print("..##..##..##..##")
        print("..##..##..##..##")
    elif n==9:
        print("##..##..##..##..##")
        print("##..##..##..##..##")
        print("..##..##..##..##..")
        print("..##..##..##..##..")
        print("##..##..##..##..##")
        print("##..##..##..##..##")
        print("..##..##..##..##..")
        print("..##..##..##..##..")
        print("##..##..##..##..##")
        print("##..##..##..##..##")
        print("..##..##..##..##..")
        print("..##..##..##..##..")
        print("##..##..##..##..##")
        print("##..##..##..##..##")
        print("..##..##..##..##..")
        print("..##..##..##..##..")
        print("##..##..##..##..##")
        print("##..##..##..##..##")
    elif n==10:
        print("##..##..##..##..##..")
        print("##..##..##..##..##..")
        print("..##..##..##..##..##")
        print("..##..##..##..##..##")
        print("##..##..##..##..##..")
        print("##..##..##..##..##..")
        print("..##..##..##..##..##")
        print("..##..##..##..##..##")
        print("##..##..##..##..##..")
        print("##..##..##..##..##..")
        print("..##..##..##..##..##")
        print("..##..##..##..##..##")
        print("##..##..##..##..##..")
        print("##..##..##..##..##..")
        print("..##..##..##..##..##")
        print("..##..##..##..##..##")
        print("##..##..##..##..##..")
        print("##..##..##..##..##..")
        print("..##..##..##..##..##")
        print("..##..##..##..##..##")
    elif n==11:
        print("##..##..##..##..##..##")
        print("##..##..##..##..##..##")
        print("..##..##..##..##..##..")
        print("..##..##..##..##..##..")
        print("##..##..##..##..##..##")
        print("##..##..##..##..##..##")
        print("..##..##..##..##..##..")
        print("..##..##..##..##..##..")
        print("##..##..##..##..##..##")
        print("##..##..##..##..##..##")
        print("..##..##..##..##..##..")
        print("..##..##..##..##..##..")
        print("##..##..##..##..##..##")
        print("##..##..##..##..##..##")
        print("..##..##..##..##..##..")
        print("..##..##..##..##..##..")
        print("##..##..##..##..##..##")
        print("##..##..##..##..##..##")
        print("..##..##..##..##..##..")
        print("..##..##..##..##..##..")
    elif n==12:
        print("##..##..##..##..##..##..")
        print("##..##..##..##..##..##..")
        print("..##..##..##..##..##..##")
        print("..##..##..##..##..##..##")
        print("##..##..##..##..##..##..")
        print("##..##..##..##..##..##..")
        print("..##..##..##..##..##..##")
        print("..##..##..##..##..##..##")
        print("##..##..##..##..##..##..")
        print("##..##..##..##..##..##..")
        print("..##..##..##..##..##..##")
        print("..##..##..##..##..##..##")
        print("##..##..##..##..##..##..")
        print("##..##..##..##..##..##..")
        print("..##..##..##..##..##..##")
        print("..##..##..##..##..##..##")
        print("##..##..##..##..##..##..")
        print("##..##..##..##..##..##..")
        print("..##..##..##..##..##..##")
        print("..##..##..##..##..##..##")
        print("##..##..##..##..##..##..")
        print("##..##..##..##..##..##..")
        print("..##..##..##..##..##..##")
        print("..##..##..##..##..##..##")
    elif n==13:
        print("##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##")
    elif n==14:
        print("##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##")
    elif n==15:
        print("##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##.##.")
        print("##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##")
    elif n==16:
        print("##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##")
    elif n==17:
        print("##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##")
    elif n==18:
        print("##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##")
    elif n==19:
        print("##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##")
    elif n==20:
        print("##..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("##..##..##..##..##..##..##..##..##..##..")
        print("..##..##..##..##..##..##..##..##..##..##")
        print("..##..##..##..##..##..##..##..##..##..##")

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3
1
2
15
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
import java.util.HashSet;
import java.util.Scanner;

public class Main {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        String s1 ="##";
        String s2 ="..";
        HashSet<Integer> set = new HashSet<>();
        set.add(1);
        set.add(2);
        for (int i = 2; i <=100 ; i++) {
            int a1 =(int)Math.pow(2,i)+1;
          set.add(a1);
            set.add(a1+1);
        }
        while (t-- > 0) {
            int n = sc.nextInt();
            for (int i = 1; i <= n*2 ; i++) {
                for (int j = 1; j <=n; j++) {
                    if(set.contains(i)){
                        if(j % 2 !=0) {
                            System.out.print(s1);
                        }else {
                            System.out.print(s2);
                        }
                    }else {
                        if(j % 2 !=0) {
                            System.out.print(s2);
                        }else {
                            System.out.print(s1);
                        }
                    }
                }
                System.out.println();
            }
        }
    }

}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios that are likely to cause the code to fail. The buggy code uses a `HashSet` to determine the pattern of the checkerboard, which is a complex and unnecessary approach. The primary issue is that the `HashSet` is incorrectly populated and used, leading to incorrect output for certain values of `n`.

### Step-by-Step Analysis:
1. **Identify the Bug**: The `HashSet` is populated with values that are not relevant to the checkerboard pattern. Specifically, it includes values like `3`, `5`, `9`, etc., which are not needed for a simple checkerboard pattern.
2. **Edge Cases**: The edge cases to consider are:
   - **Small `n` values**: `n = 1` and `n = 2` are already covered in the example.
   - **Medium `n` values**: `n = 3` and `n = 4` are also covered in the example.
   - **Large `n` values**: `n = 20` (the upper limit) to see if the code handles large inputs correctly.
   - **Values that are not in the `HashSet`**: Values like `n = 6`, `n = 10`, etc., to see if the code behaves as expected when the `HashSet` does not contain the values.

### Challenging Test Input:
We will use a test input that includes a mix of small, medium, and large values of `n`, as well as values that are not in the `HashSet` to ensure the code handles these cases correctly.

```plaintext
5
1
3
6
10
20
```

### Explanation:
- **n = 1**: This is a simple case, but it should be included to ensure the basic pattern is correct.
- **n = 3**: This is a medium-sized case that is not in the `HashSet` and should expose any issues with the pattern.
- **n = 6**: This is a larger medium-sized case that is not in the `HashSet` and should also expose any issues with the pattern.
- **n = 10**: This is a larger case that is not in the `HashSet` and should further test the code's ability to handle larger inputs.
- **n = 20**: This is the largest possible input, which should test the code's performance and correctness for the upper limit.

This test input should effectively expose the bug in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
import java.util.Scanner;

public class Personal {
    public static Scanner sc = new Scanner(System.in);

public static void main(String[] args) {
    int n = sc.nextInt();
    //int[][] arr = new int[2*n][2*n];
    int hold;
    for(int count = 0 ;count<n ; count++){
        hold = sc.nextInt();
        for(int i = 1 ;i<=2*hold ;i++){
            for(int j = 1 ;j<=(2*hold);j++){
                if(i==1 || i==2 || i==5 || i==6 || i==9 || i==10 || i==13 || i==14 || i==17 ||i==18 ){
                    if(j==1 || j==2 || j==5 || j==6 || j==9 || j==10 || j==13 || j==14 || j==17 ||j==18 ){
                        System.out.print("#");
                    }
                    else
                    System.out.print(".");
                }
                else
                {
                    if(i==3 || i==4 || i==7 || i==8 || i==11 || i==12 ||i==15 || i==16 || i==19 || i==20){
                        if(j==3 || j==4 || j==7 || j==8 || j==11 || j==12 ||j==15 || j==16 || j==19 || j==20){
                            System.out.print("#");
                        }
                        else
                        System.out.print(".");
                    }
                }

            }
            System.out.println();
        }
    }

    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on the logic used to determine the placement of '#' and '.' characters. The buggy code has a hardcoded set of conditions for specific values of `i` and `j`, which is not scalable and will fail for values of `n` that do not match these specific conditions.

### Analysis of the Buggy Code
The buggy code checks if `i` and `j` are in specific positions (1, 2, 5, 6, 9, 10, 13, 14, 17, 18) or (3, 4, 7, 8, 11, 12, 15, 16, 19, 20) to decide whether to print '#' or '.'. This approach is incorrect because it does not generalize for all values of `n`.

### Challenging Test Input
To expose this bug, we can use a value of `n` that does not match the hardcoded conditions. A good candidate is `n = 20`, which is the upper limit of the input range. This will ensure that the conditions in the buggy code will not match any of the hardcoded values, leading to incorrect output.

However, since the output length is limited, we can use a smaller value of `n` that still exposes the bug. A good choice is `n = 5`, which is large enough to show the pattern but small enough to fit within the output limit.

### Test Input
```plaintext
1
5
```

This test input will generate a `10x10` checkerboard. The buggy code will fail to correctly place the '#' and '.' characters because it does not handle the general case for `n = 5`. The expected output for `n = 5` should be:

```
##..##..##
##..##..##
..##..##..
..##..##..
##..##..##
##..##..##
..##..##..
..##..##..
##..##..##
##..##..##
```

The buggy code will produce an incorrect pattern due to its hardcoded conditions.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
#include<bits/stdc++.h>
#define ofa cin>>
#define $$ >>
#define $$$ <<
#define ll long long
#define GO ios_base::sync_with_stdio(0),cin.tie(0),cout.tie(0);
using namespace std;

// Function to print a specific number of digits after the decimal point
void PDP(double number, int Number_of_numbers) {
    cout << fixed << setprecision(Number_of_numbers) << number << endl;
}
const int nume = 3;
/*************************************************************************/
int main()
{GO
     string a = "##\n##";
     string b = "##..\n##..\n..##\n..##";
     string c = "##..##\n##..##\n..##..\n..##..\n##..##\n##..##";
     string d = "##..##..\n##..##..\n..##..##\n..##..##\n##..##..\n##..##..\n..##..##\n..##..##";

     int t ;
     ofa t ;
     while(t--)
     {
         int n ;
         ofa n ;
         if(n==1)
            cout $$$ a $$$ endl;
         else if(n==2)
            cout $$$ b $$$ endl;
         else if(n==3)
            cout $$$ c $$$ endl;
         else
            cout $$$ d $$$ endl;
     }
   return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, let's analyze the code and identify potential issues:

1. **Hardcoded Outputs**: The code uses hardcoded strings for specific values of \( n \). This means it will fail for any \( n \) other than 1, 2, 3, or 4.
2. **Input Handling**: The code reads the number of test cases and the value of \( n \) for each test case correctly.
3. **Output for Other Values of \( n \)**: The code does not handle any other values of \( n \) (i.e., \( n > 4 \) or \( n < 1 \)).

Given these points, a small-scale test input that exposes the bug should include a value of \( n \) that is not 1, 2, 3, or 4. Let's choose \( n = 5 \) as a test case. This will ensure that the code fails to produce the correct output for \( n = 5 \).

Here is the test input:

```plaintext
1
5
```

This test input will cause the code to fall into the `else` block, which prints the hardcoded string for \( n = 4 \), instead of generating the correct \( 10 \times 10 \) checkerboard for \( n = 5 \).


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
#define ll long long int
#define ld long double
#define vi vector<int>
#define vll vector<long long int>
#define vpll vector < pair <long,long> >
#define pb push_back
#define pob pop_back
#define endl "\n"

#define tt ll t; cin >> t; for(ll tt=0;tt<t;tt++)

#define f(i,e,l)  for(long long int i=e;i<l;i++)
#define ef(i,e,l) for(long long int i=e;i<=l;i++)
#define rf(i,e,l) for(long long int i=e;i>l;i--)
#define ref(i,e,l) for(long long int i=e;i>=l;i--)

#define to_upper(s) transform(s.begin(), s.end(), s.begin(), ::toupper)
#define to_lower(s) transform(s.begin(), s.end(), s.begin(), ::tolower)
// ll solve (ll n, ll k ){
//
//   }

int main (){
 ios::sync_with_stdio(0);cin.tie(0);cout.tie(0);
 tt {
  ll n;
  cin >> n;
  //if(n==1) cout << "##" << endl;
  for(ll i=1;i<=2*n;i++){
    ll x=i;
    if(x>4) x=x-4;
     for(ll j=1;j<=n;j++){
     if(x==1 || x==2){
      if(j%2!=0) cout << "##";
      else cout << "..";
     }
     else if(x==3 || x==4){
      if(j%2==0) cout << "##";
      else cout << "..";
     }

     }
     cout << endl;

  }

 }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To expose the bug in the provided buggy code, we need to focus on edge cases and scenarios that might cause the algorithm to fail. The key points to consider are:

1. **Small Values of \( n \)**: The code might handle small values of \( n \) incorrectly.
2. **Large Values of \( n \)**: The code might have issues with larger values of \( n \).
3. **Boundary Conditions**: The code might have issues at the boundaries of the checkerboard.

Given the constraints \( 1 \leq t \leq 20 \) and \( 1 \leq n \leq 20 \), we can choose a small-scale test input that includes both small and large values of \( n \) to ensure comprehensive testing.

Let's consider the following test input:
- \( t = 3 \)
- \( n = 1 \) (smallest value)
- \( n = 2 \) (a small value)
- \( n = 5 \) (a larger value)

This test input will help us identify any issues in the code for different values of \( n \).

Here is the test input:
```plaintext
3
1
2
5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
t = int(input())
for arnav_is_dumb in range(t):
  arnav_is_dumber = int(input())
  if arnav_is_dumber == 1:
    print("##")
    print("##")
  if arnav_is_dumber == 2:
    print("##..")
    print("##..")
    print("..##")
    print("..##")
  if arnav_is_dumber == 3:
    print("##..##")
    print("##..##")
    print("..##..")
    print("..##..")
    print("##..##")
    print("##..##")
  if arnav_is_dumber == 4:
    print("##..##..")
    print("##..##..")
    print("..##..##")
    print("..##..##")
    print("##..##..")
    print("##..##..")
    print("..##..##")
    print("..##..##")

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on edge cases and scenarios that the code might not handle correctly. The buggy code only handles specific values of \( n \) (1, 2, 3, and 4) and does not generalize the solution for other values. Therefore, we should test with values of \( n \) that are not explicitly handled in the code.

Here are the steps to generate a challenging test input:

1. **Test with \( n = 5 \)**: This value is not explicitly handled in the code, so it will reveal whether the code can handle a general case.
2. **Test with \( n = 20 \)**: This is the upper limit of the input range, and it will test the code's performance and correctness with the largest possible input.
3. **Test with \( n = 1 \)**: This is a basic case, but it should still be included to ensure the code handles the smallest input correctly.
4. **Test with \( n = 10 \)**: This is a medium-sized input that is not explicitly handled in the code, and it will help to see if the code can handle a moderate input size.

Given the constraints and the need for a small-scale test input, we will include three test cases: \( n = 5 \), \( n = 1 \), and \( n = 10 \).

Here is the test input:

```plaintext
3
5
1
10
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Upscaling
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given an integer $$$n$$$. Output a $$$2n \times 2n$$$ checkerboard made of $$$2 \times 2$$$ squares alternating '$$$\texttt{#}$$$' and '$$$\texttt{.}$$$', with the top-left cell being '$$$\texttt{#}$$$'.

The picture above shows the answers for $$$n=1,2,3,4$$$.

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 20$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 20$$$) — it means you need to output a checkerboard of side length $$$2n$$$.

Output: For each test case, output $$$2n$$$ lines, each containing $$$2n$$$ characters without spaces — the checkerboard, as described in the statement. Do not output empty lines between test cases.

Examples:
input:
4
1
2
3
4
output:
##
##
##..
##..
..##
..##
##..##
##..##
..##..
..##..
##..##
##..##
##..##..
##..##..
..##..##
..##..##
##..##..
##..##..
..##..##
..##..##

Note:

Buggy Code:
from collections import deque as d,Counter as cout
from itertools import combinations as comb,combinations_with_replacement as tom
'''
Deque 1
def push_front(s,n):
    s.appendleft(n)
    return s
def push_back(s,n):
    s.append(n)
    return s
def pop_front(s):
    s.popleft()
    return s
def pop_back(s):
    s.pop()
    return s
def front(s):
    return s[0]
def back(s):
    return s[-1]
def size(s):
    return len(s)
def clear(s):
    s.clear()
    return s

deq = d()
while True:
    s = input()
    g = -1
    if ' ' in s:
        g = s.find(' ')
        a = s[g:]
        s = s[:g]
        a = int(a)
    if s == 'push_front':
        deq = push_front(deq,a)
        print('ok')
    elif s == 'push_back':
        deq = push_back(deq,a)
        print('ok')
    elif s == 'pop_front':
        print(deq[0])
        deq = pop_front(deq)

    elif s == 'pop_back':
        print(deq[-1])
        deq = pop_back(deq)
    elif s == 'size':
        print(size(deq))
    elif s == 'clear':
        print('ok')
        deq.clear()
    elif s == 'back':
        print(deq[-1])
    elif s == 'front':
        print(deq[0])
    else:
        print('bye')
        exit()
'''
for _ in range(int(input())):
    a = int(input())
    p = []
    l = []
    kol = 0
    ans = 0
    flag = False
    if a == 1:
        print('##')
        print('##')

    elif a % 2 == 1:
        for i in range(a * 2+1):
            if kol < 2:
                print('##..' * (a // 2) +'##',end='')
                flag = True
                kol = kol + 1
            else:

                if ans < 2:
                    print('..##' * (a//2) +'..',sep='',end='')
                    flag = True
                    ans += 1

                else:kol = 0;ans = 0
            if flag:
                print()
                flag = False

    else:
        for i in range(a * 2+1):
            if kol < 2:
                print('##..' * (a // 2),end='')
                kol = kol + 1
                flag = True
            else:

                if ans < 2:
                    print('..##' * (a//2),sep='',end='')
                    ans += 1
                    flag = True

                else:kol = 0;ans = 0
            if flag:
                print()
                flag = False

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on edge cases and scenarios where the code might fail. Here are the key points to consider:

1. **Single Test Case (n=1):** The code has a special case for `n=1`, so it's important to test this to ensure it works correctly.
2. **Odd and Even n:** The code handles odd and even values of `n` differently. We should test both to ensure the logic is correct.
3. **Boundary Values:** Test the boundary values for `n`, such as the minimum and maximum allowed values (1 and 20).
4. **Intermediate Values:** Test some intermediate values to ensure the pattern is correctly generated.

Given these considerations, let's create a test input that includes these scenarios:

```plaintext
4
1
2
3
20
```

This test input includes:
- `n=1` to test the special case.
- `n=2` to test an even value.
- `n=3` to test an odd value.
- `n=20` to test the maximum allowed value.

This should help expose any bugs in the code's logic for generating the checkerboard pattern.