, 2026

When two standard six-sided dice are rolled, the minimum sum that
can be rolled is $1+1=2$, and so Riah must move forward at least $2$ squares on each roll. This means that Riah cannot land on the square numbered $1$, she can land on at most one of the squares numbered $11$ and $12$, and she can land on at most one of the squares numbered $16$ and $17$. To achieve the maximum score possible, the goal is to land on as many shaded squares as possible, recognizing that a choice must be made between squares $11$ and $12$, and also between squares $16$ and $17$. To begin the game, Riah cannot roll a $1$, but she can roll a $3$, thus adding $6$ points to her score. She can then roll $3$ again, adding $4$ points to her score, followed by a roll of $2$ to add $5$ more points to her score. Riah is now on the square numbered $8$, she has $6+4+5=15$ points, and she has collected the maximum number of points possible to this point in the game. From the square numbered $8$, Riah can roll a $3$ and collect $7$ points or she can roll a $4$ and collect $8$ points. Since landing on square $11$ or $12$ does not affect her ability to land on square $16$ or $17$, Riah maximizes her points by rolling $4$ and collecting $8$ points. She is now on the square numbered $12$ and has $15+8=23$ points. From the square numbered $12$, Riah can roll a $4$ and collect $10$ points or she can roll a $5$ and double her points. Since adding $10$ points gives Riah $23+10=33$ and doubling her points gives $23\times2=46$, then rolling a $5$ and landing on the square numbered $17$ maximizes Riah's points. There are no more points that can be collected before finishing the game. Thus, the greatest number of points that Riah can finish the game with is $46$.