Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n\,$, where $m\,$ and $n\,$ are relatively prime positive integers. What are the last three digits of $m+n\,$?
The probability that the $n$th flip in each game occurs and is a head is $\frac{1}{2^n}$. The first person wins if the coin lands heads on an odd numbered flip. So, the probability of the first person winning the game is $\frac{1}{2}+\frac{1}{8}+\frac{1}{32}+\cdots = \frac{\frac{1}{2}}{1-\frac{1}{4}}=\frac{2}{3}$, and the probability of the second person winning is $\frac{1}{3}$. 
Let $a_n$ be the probability that Alfred wins the $n$th game, and let $b_n$ be the probability that Bonnie wins the $n$th game. 
If Alfred wins the $n$th game, then the probability that Alfred wins the $n+1$th game is $\frac{1}{3}$. If Bonnie wins the $n$th game, then the probability that Alfred wins the $n+1$th game is $\frac{2}{3}$. 
Thus, $a_{n+1}=\frac{1}{3}a_n+\frac{2}{3}b_n$. 
Similarly, $b_{n+1}=\frac{2}{3}a_n+\frac{1}{3}b_n$. 
Since Alfred goes first in the $1$st game, $(a_1,b_1)=\left(\frac{2}{3}, \frac{1}{3}\right)$. 
Using these [recursive](https://artofproblemsolving.com/wiki/index.php/Recursion) equations: 
$(a_2,b_2)=\left(\frac{4}{9}, \frac{5}{9}\right)$
$(a_3,b_3)=\left(\frac{14}{27}, \frac{13}{27}\right)$
$(a_4,b_4)=\left(\frac{40}{81}, \frac{41}{81}\right)$
$(a_5,b_5)=\left(\frac{122}{243}, \frac{121}{243}\right)$
$(a_6,b_6)=\left(\frac{364}{729}, \frac{365}{729}\right)$
Since $a_6=\frac{364}{729}$, $m+n = 1093 \equiv \boxed{093} \pmod{1000}$.