Two players $A$ and $B$ move alternately by adding a proper divisor of $n$ to the current $n$. The goal is a number $\ge 1990$. Who wins if they started with $n=2$.

At first $A$ add 1 to 2 and form 3. Then $B$ add 1 to 3 and form 4. After that $A$ add 2 to 4 and form 6. This moves are necessary . From what is the winning players strategy to won this game.I just started this chapter and faced this critical (for me ) problem. Please help me.

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    $\begingroup$ Why can't $A$ add 1 to 4 to form 5? $\endgroup$ – Malcolm Dec 18 '17 at 3:40
  • $\begingroup$ Because I said proper divisor.But as 2&3 does not have any proper divisor so A & B add 1 $\endgroup$ – Sufaid Saleel Dec 18 '17 at 3:42
  • $\begingroup$ Is 1 not a proper divisor of 4? $\endgroup$ – Malcolm Dec 18 '17 at 3:44
  • $\begingroup$ @Malcolm a proper divisor. Any one of them. $\endgroup$ – Karn Watcharasupat Dec 18 '17 at 4:58
  • $\begingroup$ @KarnWatcharasupat The OP in the question implies that the move "add 2 to 4 and form 6" is forced. "This moves are necessary". The OP also implies this in the response to my original comment. My reference says 1 is a proper divisor of any n>1. The OP seems unclear. This needs clarification prior to considering the question. Hence my request for clarification. $\endgroup$ – Malcolm Dec 18 '17 at 14:13

Generally speaking, $1$ is regarded as a proper divisor, so there are two options when choosing a move from $4$, namely to go to $5$ or $6$.

Analyzing from the perspective of the last few moves, it's apparent that:

  • a win is immediately possible from any even number over $1327$
  • any odd number will give an even number on the next move
  • any prime number will give the subsequent number
  • an even number over $885$ not divisible by $4$ can give an odd number over $1327$

and so on, giving a trail of winning positions to try to force and losing positions to avoid.

In fact after further assessment one winning strategy is simple; you can generally win from even numbers, so always put your opponent onto an odd number (below 3/4 of the target, for safety) until you get an even number in the immediate win range.


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