Frankfurt UAS, KMITL, VGU & SPbGUT - Combinatorics & Number Theory

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2017-03-26 21:00 AKDT

## Frankfurt UAS, KMITL, VGU & SPbGUT - Combinatorics & Number Theory

#### End

2017-03-30 10:00 AKDT
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# Problem [S-Nim

Arthur and his sister Carol have been playing a game called Nim for some time now. Nim is played as follows:

• The starting position has a number of heaps, all containing some, not necessarily equal, number of beads.

• The players take turns chosing a heap and removing a positive number of beads from it.

• The first player not able to make a move, loses.

Arthur and Carol really enjoyed playing this simple game until they recently learned an easy way to always be able to find the best move:

• Xor the number of beads in the heaps in the current position (i.e., if we have $2$, $4$ and $7$ the xor-sum will be $1$ since $2\ \mathbf{xor}\ 4\ \mathbf{xor}\ 7 = 1$).

• If the xor-sum is $0$, too bad, you will lose.

• Otherwise, move such that the xor-sum becomes $0$. This is always possible.

It is quite easy to convince oneself that this works. Consider these facts:

• The player that takes the last bead wins.

• After the winning player’s last move the xor-sum will be $0$.

• The xor-sum will change after every move.

Which means that if you make sure that the xor-sum always is $0$ when you have made your move, your opponent will never be able to win, and, thus, you will win.

Understandably it is no fun to play a game when both players know how to play perfectly (ignorance is bliss). Fortunately, Arthur and Carol soon came up with a similar game, $S$-Nim, that seemed to solve this problem. Each player is now only allowed to remove a number of beads in some predefined set $S$, e.g. if we have $S = \{ 2, 5\}$ each player is only allowed to remove $2$ or $5$ beads. Now it is not always possible to make the xor-sum $0$ and, thus, the strategy above is useless. Or is it?

Your job is to write a program that determines if a position of $S$-Nim is a losing or a winning position. A position is a winning position if there is at least one move to a losing position. A position is a losing position if there are no moves to a losing position. This means, as expected, that a position with no legal moves is a losing position.

## Input

The first line of input contains a number $k$ ($1 \le k \le 100$) describing the size of $S$, followed by $k$ numbers $s_ i$ ($1 \le s_ i \le 10\, 000$) describing $S$. The second line contains a number $m$ ($1 \le m \le 100$) describing the number of positions to evaluate. The next $m$ lines each contain a number $l$ ($1 \le l \le 100$) describing the number of heaps and $l$ numbers $h_ i$ ($0 \le h_ i \le 10\, 000$) describing the number of beads in the heaps.

## Output

For each position:

• If the described position is a winning position print a ‘W’.

• If the described position is a losing position print an ‘L’.

Sample Input 1 Sample Output 1
2 2 5
3
2 5 12
3 2 4 7
4 2 3 7 12

LWW

Sample Input 2 Sample Output 2
5 1 2 3 4 5
3
2 5 12
3 2 4 7
4 2 3 7 12

WWL