A running total past a target

1. Step 1 · From a total of 9

   $\displaystyle (10 + 11 + 12 + 13) \div 4 = 11.5$

   Work back from the end. From 9, any roll finishes, so the final total is 9 plus the roll, averaged over the 4 faces.

2. Step 2 · From a total of 8

   $\displaystyle (\text{value from } 9 + 10 + 11 + 12) \div 4 = 11.125$

   A 1 takes you to 9, whose value you just found; the other rolls finish at once.

3. Step 3 · From 7 and 6

   $\displaystyle \approx 10.906 \text{ and } \approx 10.883$

   Each total’s value is the average of the values one to four steps ahead.

4. Step 4 · Back to a total of 0

   $\displaystyle \text{the same rule},\, \text{ one total at } a \text{ time},\, \text{ down to the start } = \frac{5765625}{524288}$

   Keep averaging the four totals ahead until you reach 0, the start of the game.

Answer$\displaystyle \frac{5765625}{524288} \approx 10.997$