The game of Nim (3)

This is another problem about the game of Nim, which is thoroughly
explained in
problem problem://problemsjutge.org:problems/upc/2004-semi1/5.pbm. But
here, we may have a huge number of sets, each with a huge number of
marbles. Furthermore, now we assume that the player to make the last
move loses, instead of winning.

Input

Input consists of several cases. Every case begins with the number of
sets n, followed by the number of marbles of each set, all between 0 and
10⁹. Assume 0 ≤ n ≤ 10⁵. At least one set has one or more marbles.

Output

For every case, tell if it is a winning or a losing configuration.

Hint

You should use a mathematical trick to solve this problem.

Problem information

Author: Salvador Roura

Generation: 2026-01-25T11:16:01.289Z

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