Orig Description
Score : 600 points
Problem StatementThere are N piles of stones. The i-th pile has A_i stones.
Aoki and Takahashi are about to use them to play the following game:
Starting with Aoki, the two players alternately do the following operation:
Operation: Choose one pile of stones, and remove one or more stones from it.
When a player is unable to do the operation, he loses, and the other player wins.
When the two players play optimally, there are two possibilities in this game: the player who moves first always wins, or the player who moves second always wins, only depending on the initial number of stones in each pile.
In such a situation, Takahashi, the second player to act, is trying to guarantee his win by moving at least zero and at most (A_1 - 1) stones from the 1-st pile to the 2-nd pile before the game begins.
If this is possible, print the minimum number of stones to move to guarantee his victory; otherwise, print -1 instead.
Constraints
2 \leq N \leq 300
1 \leq A_i \leq 10^{12}
InputInput is given from Standard Input in the following format:
N
A_1 \ldots A_N
OutputPrint the minimum number of stones to move to guarantee Takahashi's win; otherwise, print -1 instead.
Sample Input 12
5 3
Sample Output 11
Without moving stones, if Aoki first removes 2 stones from the 1-st pile, Takahashi cannot win in any way.
If Takahashi moves 1 stone from the 1-st pile to the 2-nd before the game begins so that both piles have 4 stones, Takahashi can always win by properly choosing his actions.
Sample Input 22
3 5
Sample Output 2-1
It is not allowed to move stones from the 2-nd pile to the 1-st.
Sample Input 33
1 1 2
Sample Output 3-1
It is not allowed to move all stones from the 1-st pile.
Sample Input 48
10 9 8 7 6 5 4 3
Sample Output 43
Sample Input 53
4294967297 8589934593 12884901890
Sample Output 51
Watch out for overflows.
Extracted Specification
An integer N (2 ≤ N ≤ 300), representing some quantity or size.
A list of integers A of size N, where each integer is between 1 and 10^12.
### Example Input:
```
2
5 3
```
### Function Signature:
Write a function f(N, A) that takes in the input.
```
def f(N: int, A: List[int]):
'''
N: an integer
A: a list of integers
'''
```
Test Case Generator
Sure, let's create a test case generator for the given problem statement. The given problem involves generating a number of piles with a specified number of stones in each pile. Here's a Python function to generate random test cases for this problem:
```python
import random
def tcgen():
# Generate a random number of piles within the constraint
N = random.randint(2, 300)
# Generate a list of N pile sizes where each pile has a random number of stones within the constraint
A = [random.randint(1, 10**12) for _ in range(N)]
return N, A
# Example of generating a test case
N, A = tcgen()
print(N)
print(" ".join(map(str, A)))
```
This function generates a random number of piles \(N\) (between 2 and 300), and for each pile, it assigns a random number of stones \(A[i]\) (between 1 and \(10^{12}\)).
You can use this function to produce random test cases for your problem. Each time you call `tcgen()`, it will produce a new, random test case that adheres to the given constraints, ensuring a variety of test cases to validate your solution.
Here's an example of how to use the generated test cases:
```python
# Example usage of the generated test case
N, A = tcgen()
print("Generated Test Case:")
print(f"N: {N}")
print(f"A: {A}")
```
This will print out the generated values for \(N\) and \(A\), which you can then use to test your solution to the problem.
Extract Arguments
def extract_arguments(fh):
N = int(fh.readline().strip())
A = list(map(int, fh.readline().strip().split()))
return N, A