Since the roll-over amount is formula_2, chopsticks is a base-formula_2 game. Each position is formula_27 digits long. Enumerating all numbers in base-formula_2 with formula_27 digits gives us formula_22 positions. However, most of these positions are incorrect notations (e.g. 001210, 010120, and 100021 for formula_31). They appear different but are functionally the same in gameplay. To find the number of functionally distinct positions, we square the number of functionally distinct pairs. For a roll-over of formula_2 and formula_6 hands, there are formula_34 distinct pairs, where formula_34 is the formula_2-th formula_6-simplex number. Since either player could have any of these pairs, we simply square the resulting value, which gives us formula_23 functionally distinct positions.