Question - Need your help!

Someone will finish the game when the remainder is 1, 2, 3, 4, 5 (remaining number of sticks).

This basicallly means the number of sticks withdrawn in the game so far should leave a remainder R between 1 and 5 (inclusive)

R mod 6n = 1, 2, 3, 4, 5 (n = 1, 2, 3 ....)

The above statement implies all multiples of 6 will lead to a win. The catch is to make sure (rita + sam) mod 6 is zero for sam to win.

Because the player can only remove up to 5 sticks and must remove at least 1, if the number of sticks is a multiple of 6, the second player (Sam) can control the game, and if he understands this, he can win the game every time. This doesn't necessarily meant they will win every time, as was pointed out in a prior example. But if he is smart he can eliminate rounds of 6 so that in the final turns there are 6 left for Rita and Sam wins no matter what.

You can think of it like this: if there were 6 sticks on the table. Sam would win no matter what (if he chose right). This is because no matter what Rita chooses (1-5) there are just enough sticks so that Sam can pick them all up. No matter how many sticks they start with sam can eliminate 6-whatever Rita chose, so that in the final rounds Rita is left with 6.

Take, for instance, this example starting with 18 sticks:

-Rita takes 1

-Sam takes 6-1 or 5 (so one round of 6 is eliminated)

-Rita takes 4

-Sam takes 6-4 or 2 (another round of 5 is eliminated)

-Rita is left with 6, no matter what she chooses, Sam can take the rest and finish the game.

I hope this helps!