问题

### Instructions: You are an expert Python programmer. You will be given a question (problem specification) and will generate a correct Python program that matches the specification and passes all tests. You will NOT return anything except for the program.
### Question:
You are given two positive integers x and y.
In one operation, you can do one of the four following operations:

Divide x by 11 if x is a multiple of 11.
Divide x by 5 if x is a multiple of 5.
Decrement x by 1.
Increment x by 1.

Return the minimum number of operations required to make x and y equal.

Example 1:

Input: x = 26, y = 1
Output: 3
Explanation: We can make 26 equal to 1 by applying the following operations:
1. Decrement x by 1
2. Divide x by 5
3. Divide x by 5
It can be shown that 3 is the minimum number of operations required to make 26 equal to 1.

Example 2:

Input: x = 54, y = 2
Output: 4
Explanation: We can make 54 equal to 2 by applying the following operations:
1. Increment x by 1
2. Divide x by 11
3. Divide x by 5
4. Increment x by 1
It can be shown that 4 is the minimum number of operations required to make 54 equal to 2.

Example 3:

Input: x = 25, y = 30
Output: 5
Explanation: We can make 25 equal to 30 by applying the following operations:
1. Increment x by 1
2. Increment x by 1
3. Increment x by 1
4. Increment x by 1
5. Increment x by 1
It can be shown that 5 is the minimum number of operations required to make 25 equal to 30.


Constraints:

1 <= x, y <= 10^4

### Format: You will use the following starter code to write the solution to the problem and enclose your code within delimiters.
```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:

```

### Answer: (use the provided format with backticks)

答案

模型 模型答案 正确答案
金声玉亮2.0 ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
from collections import deque

def bfs(start, target):
queue = deque([(start, 0)])
visited = set()

while queue:
current, steps = queue.popleft()

if current == target:
return steps

if current in visited:
continue

visited.add(current)

if current % 11 == 0:
queue.append((current // 11, steps + 1))
if current % 5 == 0:
queue.append((current // 5, steps + 1))
queue.append((current - 1, steps + 1))
queue.append((current + 1, steps + 1))

return -1

return bfs(x, y)
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
deepseek-r1

To solve this problem, we need to determine the minimum number of operations required to make two positive integers x and y equal using specific operations. The operations allowed are dividing x by 11 or 5 (if possible), incrementing x by 1, or decrementing x by 1.

### Approach
1. **Immediate Check for y >= x**: If y is greater than or equal to x, the minimum operations required are simply the difference y - x since we can only increment x to reach y.
2. **Breadth-First Search (BFS) for y < x**: When y is smaller than x, we use BFS to explore all possible states starting from x. This allows us to efficiently find the shortest path (minimum operations) to reach y by considering all valid operations at each step. BFS is chosen because it guarantees the shortest path in an unweighted graph, which is suitable here as each operation can be considered as moving to a new state.

### Solution Code
```python
from collections import deque

class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
if x == y:
return 0
if y >= x:
return y - x

visited = set()
q = deque()
q.append((x, 0))
visited.add(x)

while q:
current_x, steps = q.popleft()

next_states = []
if current_x % 5 == 0:
next_states.append(current_x // 5)
if current_x % 11 == 0:
next_states.append(current_x // 11)
next_states.append(current_x + 1)
next_states.append(current_x - 1)

for next_x in next_states:
if next_x == y:
return steps + 1
if next_x >= 0 and next_x not in visited:
if next_x <= 100000: # Prevent excessively large values
visited.add(next_x)
q.append((next_x, steps + 1))

return -1 # According to problem constraints, this line is unreachable
```

### Explanation
1. **Immediate Check**: If y is greater than or equal to x, we directly return the difference since we can only increment x.
2. **BFS Initialization**: We initialize a queue with the starting value x and 0 steps. We also maintain a visited set to avoid revisiting the same state.
3. **State Exploration**: For each current state (value of x and steps taken), we generate all possible next states by applying the allowed operations. These include division by 5 or 11 (if possible), increment, and decrement.
4. **Check for Target**: If any next state matches y, we return the steps taken plus one (current step). If not, we add valid next states (non-negative and not visited) to the queue for further exploration.
5. **Boundary Check**: To prevent processing excessively large values, we cap the next state at 100000. This is a heuristic to balance between practical limits and ensuring correctness.

This approach efficiently explores all possible paths using BFS, ensuring the shortest path (minimum operations) is found when y is less than x.		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
gpto1 ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
from collections import deque

max_limit = 10000
visited = set()
queue = deque()
queue.append((x, 0)) # (current_value, number of operations)
visited.add(x)

while queue:
current_x, ops = queue.popleft()
if current_x == y:
return ops
# Generate possible next moves
possible_next = []
if current_x % 11 == 0:
next_x = current_x // 11
if 1 <= next_x <= max_limit and next_x not in visited:
visited.add(next_x)
queue.append((next_x, ops +1))
if current_x % 5 == 0:
next_x = current_x // 5
if 1 <= next_x <= max_limit and next_x not in visited:
visited.add(next_x)
queue.append((next_x, ops +1))
# Decrement x by 1
next_x = current_x -1
if 1 <= next_x <= max_limit and next_x not in visited:
visited.add(next_x)
queue.append((next_x, ops +1))
# Increment x by 1
next_x = current_x +1
if 1 <= next_x <= max_limit and next_x not in visited:
visited.add(next_x)
queue.append((next_x, ops +1))
# If y is not reachable
return -1
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
gpt-4o-2024-08-06 ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
from collections import deque

# BFS approach to find the minimum number of operations
queue = deque([(x, 0)]) # (current value of x, number of operations)
visited = set()

while queue:
current, ops = queue.popleft()

if current == y:
return ops

if current in visited:
continue

visited.add(current)

# Try all possible operations
if current % 11 == 0:
queue.append((current // 11, ops + 1))
if current % 5 == 0:
queue.append((current // 5, ops + 1))

# Decrement and increment operations
queue.append((current - 1, ops + 1))
queue.append((current + 1, ops + 1))

return -1 # This should never be reached given the problem constraints
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
Qwen2-72B-Instruct ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
q = deque([(x, 0)])
visited = set([x])
while q:
cur, step = q.popleft()
if cur == y:
return step
for next in [cur-1, cur+1, cur//5, cur//11]:
if next > 0 and next not in visited:
visited.add(next)
q.append((next, step+1))
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
DeepSeek-Coder-V2-Lite-Instruct ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
def operations_needed(a, b):
ops = 0
while a != b:
if a < b:
if a % 11 == 0 and b % 11 == 0 and a // 11 >= (b - a) // 5:
ops += (b - a) // 5
a += ((b - a) // 5) * 5
elif a % 5 == 0 and b % 5 == 0 and a // 5 >= (b - a):
ops += (b - a)
a += (b - a)
elif a < b:
a += 1
ops += 1
else:
a -= 1
ops += 1
else:
if a % 11 == 0 and b % 11 == 0 and (a - b) // 11 <= (a - b) // 5:
ops += (a - b) // 5
a -= ((a - b) // 5) * 5
elif a % 5 == 0 and b % 5 == 0 and (a - b) // 5 <= (a - b):
ops += (a - b)
a -= (a - b)
elif a > b:
a -= 1
ops += 1
else:
a += 1
ops += 1
return ops

return min(operations_needed(x, y), operations_needed(y, x))

# Example usage:
# sol = Solution()
# print(sol.minimumOperationsToMakeEqual(26, 1)) # Output: 3
# print(sol.minimumOperationsToMakeEqual(54, 2)) # Output: 4
# print(sol.minimumOperationsToMakeEqual(25, 30)) # Output: 5
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
Meta-Llama-3.1-405B-Instruct-Turbo ```python
from collections import deque

class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
queue = deque([(x, 0)])
visited = set([x])

while queue:
curr_x, steps = queue.popleft()

if curr_x == y:
return steps

for next_x in [curr_x - 1, curr_x + 1, curr_x // 5 if curr_x % 5 == 0 else None, curr_x // 11 if curr_x % 11 == 0 else None]:
if next_x is not None and next_x not in visited:
queue.append((next_x, steps + 1))
visited.add(next_x)
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
Meta-Llama-3.1-70B-Instruct-Turbo ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
if x == y:
return 0
elif x < y:
return self.minimumOperationsToMakeEqual(y, x)

diff = x - y
ops = diff // 11 + diff % 11 // 5 + diff % 11 % 5
return ops
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
baichuan4 ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
# Calculate the absolute difference between x difference = abs(x - y)

# Initialize the count of operations to 0
operations = 0

# While the difference is greater than 0, apply the most efficient operation
while difference > 0:
if difference % 11 == 0:
difference //= 11
operations += 1
elif difference % 5 == 0:
difference //= 5
operations += 1
elif difference > 1:
difference -= 1
operations += 1
else:
difference += 1
operations += 1

return operations
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
doubao ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
count = 0
while x!= y:
if x > y:
if x % 11 == 0:
x //= 11
elif x % 5 == 0:
x //= 5
else:
x -= 1
else:
if y % 11 == 0 and x < y // 11:
y //= 11
elif y % 5 == 0 and x < y // 5:
y //= 5
else:
x += 1
count += 1
return count
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
gpto1 ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
from collections import deque

max_limit = 10000
visited = set()
queue = deque()
queue.append((x, 0)) # (current_value, number of operations)
visited.add(x)

while queue:
current_x, ops = queue.popleft()
if current_x == y:
return ops
# Generate possible next moves
possible_next = []
if current_x % 11 == 0:
next_x = current_x // 11
if 1 <= next_x <= max_limit and next_x not in visited:
visited.add(next_x)
queue.append((next_x, ops +1))
if current_x % 5 == 0:
next_x = current_x // 5
if 1 <= next_x <= max_limit and next_x not in visited:
visited.add(next_x)
queue.append((next_x, ops +1))
# Decrement x by 1
next_x = current_x -1
if 1 <= next_x <= max_limit and next_x not in visited:
visited.add(next_x)
queue.append((next_x, ops +1))
# Increment x by 1
next_x = current_x +1
if 1 <= next_x <= max_limit and next_x not in visited:
visited.add(next_x)
queue.append((next_x, ops +1))
# If y is not reachable
return -1
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
kimi ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
def helper(x):
operations = 0
while x % 11 == 0:
x //= 11
operations += 1
while x % 5 == 0:
x //= 5
operations += 1
return operations + abs(x)
return min(helper(x), helper(y))
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
minimax ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
# Initialize the number of operations to 0
operations = 0

# Make x and y equal by incrementing or decrementing x
while x != y:
# If x is greater than y, decrement x
if x > y:
x -= 1
# If x is less than y, increment x
else:
x += 1
# Increment the number of operations
operations += 1

# Return the total number of operations performed
return operations
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
wenxinyiyan ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
# Define a helper function to calculate the minimum operations for a given x and y
def min_operations(a: int, b: int) -> int:
# Base case: if a and b are equal, return 0 operations
if a == b:
return 0

# Initialize the minimum operations to a large value
min_ops = float('inf')

# Try all four possible operations on a and update the minimum operations
if a % 11 == 0:
min_ops = min(min_ops, 1 + min_operations(a // 11, b))
if a % 5 == 0:
min_ops = min(min_ops, 1 + min_operations(a // 5, b))
min_ops = min(min_ops, 1 + min_operations(a - 1, b))
min_ops = min(min_ops, 1 + min_operations(a + 1, b))

return min_ops

# Call the helper function with the given x and y
return min_operations(x, y)
```
This solution uses a recursive helper function `min_operations` to calculate the minimum number of operations required to make `x` and `y` equal. It explores all possible operations on `x` (divide by 11, divide by 5, decrement by 1, increment by 1) and recursively calls itself with the updated value of `x` and the original value of `y`. The base case is when `x` and `y` are equal, in which case it returns 0 operations. The minimum number of operations is tracked using the variable `min_ops` and updated after each operation is tried. Finally, the solution calls the helper function with the given `x` and `y` and returns the result.		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
zhipu ```python
class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
from collections import deque

if x == y:
return 0

visited = set()
queue = deque([(x, 0)])

while queue:
current, steps = queue.popleft()

if current == y:
return steps

if current in visited:
continue

visited.add(current)

if current % 11 == 0:
queue.append((current // 11, steps + 1))
if current % 5 == 0:
queue.append((current // 5, steps + 1))
queue.append((current - 1, steps + 1))
queue.append((current + 1, steps + 1))

return -1 # This line should theoretically never be reached
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]
deepseek-v3 ```python
from collections import deque

class Solution:
def minimumOperationsToMakeEqual(self, x: int, y: int) -> int:
if x == y:
return 0

queue = deque()
queue.append((x, 0))
visited = set()
visited.add(x)

while queue:
current, steps = queue.popleft()

# Check possible operations
if current % 11 == 0:
next_val = current // 11
if next_val == y:
return steps + 1
if next_val not in visited:
visited.add(next_val)
queue.append((next_val, steps + 1))

if current % 5 == 0:
next_val = current // 5
if next_val == y:
return steps + 1
if next_val not in visited:
visited.add(next_val)
queue.append((next_val, steps + 1))

next_val = current - 1
if next_val == y:
return steps + 1
if next_val not in visited and next_val >= 1:
visited.add(next_val)
queue.append((next_val, steps + 1))

next_val = current + 1
if next_val == y:
return steps + 1
if next_val not in visited and next_val <= 10**4:
visited.add(next_val)
queue.append((next_val, steps + 1))

return -1 # Should not reach here as per constraints
```		 [{"input": "26\n1", "output": "3", "testtype": "functional"}, {"input": "54\n2", "output": "4", "testtype": "functional"}, {"input": "25\n30", "output": "5", "testtype": "functional"}]