game theory

I think there are many ways for A to win.A simple way is A turning over 5 first , as follows :
Case 1 :
After the 1st move of B , the remains is not the multiple of 7 , then A
can make remains is the multiple of 7 after each move to win since
(31 - 5) / 7 < 4 .

Example :A : 5 ........... Remain 26B : 1 ........... Remain 25
A : 4 ........... Remain 21 = 7 x 3B : 5 ........... Remain 16
A : 2 ........... Remain 14 = 7 x 2B : 5........... Remain 9
A : 2 ...........Remain 7 = 7 x 1B : 3 ........... Remain 4
A : 4 ........... Remain 0 = 7 x 0 , A win ! Case 2 :
After the 1st move of B , the remains is the multiple of 7 (then B must make remains is the multiple of 7 any time otherwise A can make it the multiple of 7 to win) , but A can make B use '5' up to win :
A : 5 ...........Remain 26 = 21 + 5
B : 5 ........... Remain 21 = 7 x 3A : 2 ........... Remain 19 = 14 + 5
B : 5 ........... Remain 14 = 7 x 2A : 2 ........... Remain 12 = 7 + 5
B : 5 ........... Remain 7 = 7 x 1
(5 is used up) A : 2 ........... Remain 5
B : No 5 to move A win !

2012-02-11 13:49:17 補充：
002 :

Not only case 1 , you can say case 2 also would not be happened since B always lose in both cases.
I just to list out both cases for reference.

2012-02-11 14:05:39 補充：
Another way for A to win :

A : 1 ...... Remain 30
B : 2 ...... Remain 28 = 7 x 4

A : 1 ...... Remain 27
B : 6 ...... Remain 21 = 7 x 3

A : 1 ..... Remain 20
B : 6 ..... Remain 14 = 7 x 2

A : 1 .... Remain 13
B : 6 .....Remain 7 = 7 x 1

A : 6 ... Remain 1
B : No 1 to move ,

A win!

Sorry, I think ☂雨後陽光☀ is wrong ...

There is only 1 way to win: Turn 5 at the start.

The critical point of this game is 3,10,17,24,31, which comes from minus 7 start from 31, player would absolutely make these points in his turn. So the obvious way should be turn 3 at the start, but there is 1 ways that B can win if A turn 3 at the start, as there are only 4 cards of each type.

i.e.:
1st round

A B A Sum 3 4 3 10



2nd round

A B A Sum (10) 4 3 17



3rd round

A B A Sum (17) 4 3 24



4th round

A B A Sum (24) 4 no3 Any but not 31

The exceptional case is 5, because there are only 4 card of 5s. As B would keep the number in 10,17,24,31 in his turn, the game should be:

1st round


A B Sum
5 5 10




2nd round


Start A B Sum 10 2 5 17



3rd round


Start A B Sum 17 2 5 24



4th round


Start A B Sum 24 2 1 27 24
2 28 24
3 29 24
4 30 24
no 5




5th round


Start A Sum
27 4 31
28 3 31
29 2 31
31 1 31


As B would absolutely keep his turn in 10,17,24,31 point, case 1 of ☂雨後陽光☀ would not be happened.

So the conclusion is there is only 1 way to win: turn 5 at the start.