Lucky number 23

If Jeff wants to ensure that he will be the player who says $23$ he will need to force his opponent to say one of the numbers $19, 20, 21$ or $22$ . To ensure that this happens, it is necessary for Jeff to say $18$ . This will happen if Jeff forces his opponent to say one of $14, 15, 16$ or $17$ . The pattern here is that if Jeff wants to guarantee saying a number $n$ it will be necessary for him to first say the number $n - 5$ . Thus in turn he will need to have said the numbers $13, 8$ , and to start with, $3$ .

If Jeff were to start with $1$ or $2$ then his opponent could say $3$ and then move up the "chain" as discussed above. If he were to start with $4$ then his opponent would say $8$ and then follow the winning chain. Thus only by first saying $\boxed{3}$ can he guarantee a win.

Players can only say $1,2,3,4$ .If first player always need to win then there must be odd number of turns.

Here $1+4=2+3=5$ then if block of $5's$ can be used to reach $23$ i.e. $23=5+5+5+5+3$ .There will be $4$ blocks of fives.Fives can be created as shown above.So every five will take two turns.Total $4*2=8$ turns and last turn for $3$ ,total of $9$ turns.So,if first player starts with saying $3$ then there is a guaranteed win.

The first player will say $3$ .then second player will say something among $1,2,3,4$ (say $r$ ) then again first player should say $5-r$ .In this way the game continues and first player will win.

It is neccesary to say $3$ to win the game and also if you follow the above algorithm then you are an winner always.