Two persons play a game with matches. Once at a time they take one, three or four matches out of the box but not two or more than four. The one who takes the last match is the winner. There are 50 matches in the box. Is there a strategy than guarantees a win if you get to draw first. Which is it?
Good luck!
How about this approach:
Let N = number of matches that are left in the box. I take 1 match on the first turn, and from then on I always take a quantity of matches that will leave my opponent with either N modulo 7 = 0 or N modulo 7 = 2 matches. In other words, after each turn I want to leave him with one of the following conditions:
0
2
7
9
14
16
21
23
28
30
35
37
42
44
49
If I take 1 on the first move, that leaves him with 49. No matter whether he takes 1, 3 or 4 I can always take an appropriate number on my next turn to get him either to 42 (which evenly divides by 7) or 44 (which divides by 7 with remainder 2). If I continue with this strategy my opponent will ultimately end up with either 0 (in which case I've already won), or 2, in which case I win because his only option is to take 1, leaving me to take 1 for the win.
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