Time Limit: 10 Seconds
Memory Limit: 32768 KB
Christiane and Matthias are playing a new game, the Number Game. The rules of
the Number Game are:
Christian and Matthias take turns in choosing integer numbers greater than or
equal to 2. The following rules restrict the set of numbers which may be chosen:
R1 A number which has already been chosen by one of the players or a multiple
of such a number cannot be chosen. (A number z is a multiple of a number y if
z can be written as y * x and x is a positive integer.)
R2 A sum of two such multiples cannot be chosen either.
R3 For simplicity, a number which is greater than 20 cannot be chosen either.
This enables a lot more NPCs (Non-Personal-Computers) to play this game.
The player who cannot choose any number anymore looses the Number Game.
Here is an example: Matthias starts by choosing 4. Then Christiane is not allowed
to choose 4, 8, 12, etc. Let us assume her move is 3. Now, the numbers 3, 6,
9, etc. are excluded, too; furthermore, numbers like: 7 = 3 + 4, 10 = 2 * 3
+ 4, 11 = 3 + 2 * 4, 13 = 3 * 3 + 4, . . . are not available. So, in fact, the
only numbers left are 2 and 5. Matthias now says 2. Since 5 = 2 + 3 is now forbidden,
too, he wins because there is no number for Christiane's move left.
Your task is to write a program which will help to play the Number Game. In
general, i.e., without rule R3, this game may go on forever. However, with rule
R3, it is possible to write a program that finds a strategy to win the game.
Given a game situation (a list of numbers which are not yet forbidden), your
program should output all winning moves. A winning move is a move by which the
player whose turn it is can force a win no matter what the other player will
do. Now we define these terms more formally:
A loosing position is a position in which either
1. all numbers are forbidden, or
2. no winning move exists.
A winning position is a position in which a winning move exists.
A winning move is a move after which the position is a loosing position.
The first line contains the number of scenarios.
The input for each scenario describes a game position. It begins with a line
containing the number a, 0 <= a < 20 of numbers which are still available.
Next follows a single line with the a numbers still available, separated by
You may assume that all game positions in the input could really occur in the
Number Game (for example, if 3 is not in the list of numbers available, 6 will
not be, either).
The output for each scenario begins with a line containing "Scenario #i:"
where i is the number of the scenario starting at 1. In the next line either
print "There is no winning move." if this is true for the position
of the current scenario, or "The winning moves are: w1 w2 . . . wk."
where the wi are all the winning moves, in ascending order, separated by single
blanks. The output for each scenario should be followed by a blank line.
The winning moves are: 2.
There is no winning move.
Source: Mid-Central European Regional Contest 2000