A strip of 41 squares is numbered from left to right and a token is placed on the square marked . Pinar rolls a pair of standard six-sided dice and moves the token right a number of squares equal to the total of the dice roll. If Pinar rolls doubles, then she rolls the dice a second time and moves the token in the same manner. If Pinar gets doubles again, she rolls the dice a third time and moves the token in the same manner. If Pinar rolls doubles a third time she simply moves the token to the square marked The expected value of the square that the token ends on can be expressed as where and are coprime positive integers. What is the value of
Details and assumptions
Rolling doubles occurs when the two dice each have the same number showing.
Regardless of the outcome of the first roll, Pinar "moves the token right a number of square equal to the total of the dice roll".