In the Beast Ball league, if a player loses a match, they gain $1$ point for participating. If a player instead wins, they gain $3$ points if they also won their previous game, otherwise, they gain $2$ points for this victory. Grogg enters the league with $0$ points. Find the number of different sequences of wins and loses that will give Grogg $10$ points. (If a player wins their first match, they get $2$ points. They did not win their previous match since they didn't have one.)
My understanding of this question is that it is very similar to the stock question of climbing a flight of stairs with $10$ steps while only being able to climb $1$, $2$, or $3$ steps at a time. For this question, however, you cannot follow a $1$ with a $2$, $2$ with a $2$, or a $3$ with a $2$. How can I eliminate those possibilities?
I would just count by hand. Start with all wins and discover that you can't get there because three gives $8$ points and four gives $11$, so you can' start with more than three wins. Now backtrack. You find $WWWLL$ gives $10$ points, which is the only way starting from $WWW$. If you start with $WWL$ you have six so far, so can have four $L$s or two $L$s and a $W$ in some order. Keep going.
If we change the target to some larger number like $50$ so that hand counting is impractical I would make two coupled recurrences. Define $A(n)$ as the number of ways to score $n$ points ending with a loss and $B(n)$ as the number of ways to score $n$ points ending with a win. You have $A(n)=A(n-1)+B(n-1)$ because a loss always adds one point. What is the recurrence for $B(n)$? Now build a spreadsheet with a column for $n$, a column for $A(n)$, and a column for $B(n)$, write the recurrence formulas in one row and copy down.