We start with finitely many 1 's and possibly some 0 's in between. Then each entry in the other rows is obtained from the Base 2 sum of the two numbers diagonally above it in the preceding row. We may formulate the game as follows: Define d 1 , j recursively for 1 , a non-negative integer, and j an arbitrary integer by the rules: d 0 , j = { 1           for       j = 0 , k                   ( I ) 0       or       1       for       0 < j < k k\left( {II} \right)$" display="block"> d 0 , j = 0       for       j < 0       or       j > k                             ( I I ) d i + 1 , j = d i , j + 1 ( mod 2 )       for       i ≥ 0.             ( I I I ) Now, if we interpret the number of 1 's in row i as the coefficient a i of a formal power series, then we obtain a growth function, f ( x ) = ∑ i = 0 ∞ a i x i . It is interesting that there are cases for which this growth function factors into an infinite product of polynomials. Furthermore, we shall show that this power series never represents a rational function.