They decided the following game is suitable to the occasion:
taking alternating turns they say numbers, keeping the following rules:
- In the first turn, Haydn says the number $2$.
- After that, the player on turn says a number which is either the sum or the product of two (not necessarily distinct) previously appeared numbers.
- The number which he says should differ from all the previously mentioned ones, and cannot be greater than $1756$.
For clarification: as a second move, Beethoven can say only $4$, which can be composed as $2+2$ or $2\times2$. After that, Haydn has a real choice answering either $6$ ($2+4$), $8$ ($4+4$ or $2\times4$), or $16$ ($4\times4$).
The winner is the one that says $1756$ - the birthyear of Mozart.
Which of the two players has a winning strategy?
I'm not the original author of this puzzle. It was a problem more than 10 years ago at a mathematics competition for Hungarian high school students.
Furthermore, I'll reward an additional 50-point bounty for a general answer, that is determining the winning player and strategy if the target number (and also the upper limit of numbers) is $n$.