Not to Reach 10000

We show that Dan wins. We will correctly identify the "Kernels" in this combinatorial game.

Note that if Dan can plan a strategy such that he can bag the number $9999$ , then we are done. For this, he will plan to say the number $99$ ; Sam must then say a number in $[100,9800]$ , after which Dan can always choose $9999$ .

In turn, $99$ can be stolen if $9$ can be stolen, which in turns requires that Dan says the number $3$ . But Dan can do this, so we are done.

So Dan's strategy would be to play the numbers $1,3,9,99,9999$ in that order.

If any player chooses number $x$ , the other player can choose any number between $x^2-1$ & $x+1$ . Irrespective of the choice the 1st player can always choose numbers:- $x^2$ to $x^2+2x$ . This means that if we 'need' a number which is between $x^2$ & $x^2+2x$ , we need $x$ .

Thus, if we need any given number, all we need is its square root rounded down!

Here we need the number 9999 (to win), whose square root is slightly less 100, Thus we need 99. Square root of 99 is slightly less than 10 so we need 9. square root of 9 is 3 so we need 3. 3 is chosen by Dan. So Dan will win the round by choosing the numbers in reverse order i.e 3,9,99,9999