Arbitrary Nim

AtCoder

IDarc168_b

Time2000ms

Memory256MB

Difficulty—

There are $N$ piles of stones, and the $i$\-th pile ($1 \leq i \leq N$) has $A_i$ stones. You will choose a positive integer $k$. Then, Alice and Bob will play a game with these piles as follows. * Starting with Alice, the players take turns to play. * In each turn, the player must choose a non-empty pile and remove from it some number of stones between $1$ and $k$, inclusive. The person who cannot make a move on their turn loses, and the other person wins. You want to choose a positive integer $k$ for which there is a winning strategy for Alice. Determine if such a $k$ exists. If it exists, determine if there is a maximum value for such $k$. If the maximum value exists, provide that value. ## Constraints * $1 \leq N \leq 250000$ * $1 \leq A_i \leq 10^9$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\cdots$ $A_N$ [samples]

Samples

Input #1

3
1 2 3

Output #1

2

For example, if $k=2$, Alice has a winning strategy. If $k \geq 3$, Alice has no winning strategy, so the answer is $k=2$.

Input #2

4
1 2 3 4

Output #2

\-1

For example, Alice has winning strategies for all $k=100,200,300,\cdots$. Thus, there is no maximum value for $k$, so print $-1$.

Input #3

2
100 100

Output #3

0

No matter what $k$ is chosen, Alice has no winning strategy. Thus, print $0$.

API Response (JSON)

{
  "problem": {
    "name": "Arbitrary Nim",
    "description": {
      "content": "There are $N$ piles of stones, and the $i$\\-th pile ($1 \\leq i \\leq N$) has $A_i$ stones. You will choose a positive integer $k$. Then, Alice and Bob will play a game with these piles as follows. *  ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc168_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ piles of stones, and the $i$\\-th pile ($1 \\leq i \\leq N$) has $A_i$ stones.\nYou will choose a positive integer $k$. Then, Alice and Bob will play a game with these piles as follows.\n\n*  ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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