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u probably meant numbers 1-9 and not "greater than 10" meaning 1-10? Anyway it doesn't change much ) If both players are smart enough, the game has no sence as you can say in the beginning who wins.

So.. the winner is the one whose turn it is when the sum is 90-99, it means if the loser got the sum 89, he has no chances to win.. so the winner has to make 89. You can make 89 out of 79-88, so the one who got 78 is the loser and so on (89, 78, 67, 56, 45, 34, 23, 12, 1 are the winner numbers)...
Using some mathematical terms you count (x_min + x_max), it is an interval where you can manipulate your opponent's numbers (doesn't matter what your opponent takes, you can always reach your next winner number). In our case it's 1+10=11. So the one who gets the winner number first is the winner :) All he has to do to get to the next winner number is to name a number (11-x_i) where x_i is the number of the opponent. In our case the first player has to take 100mod11=1 to win, if he doesn't, the second one can win if he gets 12. If you want more math, make a sequence out of this ;D

P.S. with numbers 1-9 the winner numbers are 10, 20, 30, .... So the winner out of two smart guys is the second one ;)

Ell, you're right. Thanks for the answer!