Problem: There 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.

Submission attempts (in order):
# AFTER EDITORIAL


def solve(ls):
    import math

    x, y, *ls = ls
    z = x + y
    s = 0
    for a in ls:
        s ^= a

    #
    # Max a <= x  s.t.
    #   0 = NimSum(a, (x + y) - a, ls)
    # <=>
    # Max a <= x s.t.
    #   ∃b.
    
    #     a ^ b = NimSum(ls)
    # <=>
    # Max a <= x s.t.
    #   ∃b.
    
    #     a ^ b = NimSum(ls)
    #

    #
    # Thus it amounts to solve
    #   a ^ b = s
    
    #
    # which is bit-wise independent as
    #   a  b  ^  &
    #   0  0  0  0
    #   0  1  1  0
    #   1  0  1  0
    #   1  1  0  1
    #

    if (z - s) < 0 or (z - s) % 2 == 1:
        return -1

    t = (z - s) // 2
    s_t = s & t

    if s_t > 0 or t > x:
        return -1

    a = t
    b = t
    first_set = math.floor(math.log2(s))
    for i in range(first_set + 1)[::-1]:
        k = 1 << i
        if s & k:
            if (a | k) <= x:  # Greedy is optimal
                a = a | k
            else:
                b = b | k
    return x - a


def main(istr, ostr):
    n = int(istr.readline().strip())
    ls = list(map(int, istr.readline().strip().split()))
    result = solve(ls)
    print(result, file=ostr)


if __name__ == "__main__":
    import sys

    main(sys.stdin, sys.stdout)


# AFTER EDITORIAL


def solve(ls):
    import math

    x, y, *ls = ls
    z = x + y
    s = 0
    for a in ls:
        s ^= a

    #
    # Max a <= x  s.t.
    #   0 = NimSum(a, (x + y) - a, ls)
    # <=>
    # Max a <= x s.t.
    #   ∃b.
    
    #     a ^ b = NimSum(ls)
    # <=>
    # Max a <= x s.t.
    #   ∃b.
    
    #     a ^ b = NimSum(ls)
    #

    #
    # Thus it amounts to solve
    #   a ^ b = s
    
    #
    # which is bit-wise independent as
    #   a  b  ^  &
    #   0  0  0  0
    #   0  1  1  0
    #   1  0  1  0
    #   1  1  0  1
    #

    if (z - s) < 0 or (z - s) % 2 == 1:
        return -1

    t = (z - s) // 2
    s_t = s & t

    if s_t > 0 or t > x:
        return -1

    a = t
    b = t
    first_set = math.floor(math.log2(s + 1))
    for i in range(first_set + 1)[::-1]:
        k = 1 << i
        if s & k:
            if (a | k) <= x:  # Greedy is optimal
                a = a | k
            else:
                b = b | k
    if a == 0:
        return -1
    return x - a


def main(istr, ostr):
    n = int(istr.readline().strip())
    ls = list(map(int, istr.readline().strip().split()))
    result = solve(ls)
    print(result, file=ostr)


if __name__ == "__main__":
    import sys

    main(sys.stdin, sys.stdout)


Verdicts: Runtime Error

Accepted