Asume you can build a self replicating machine called X. X can perform two functions, further build another machine X in 24 hours or generate a product Y, also in 24 hours. In a 30 day, calculate the maximum number of product Y, given infinite resources? What is the strategy and maths involved?
The question arises because of watching a video on use of Von Neuman technology to terraform planets. Thanks.
As you correctly found out, the solution is to replicate the $X$ machines until the last day and then produce $$ 2^{d-1} $$ copies of machine $Y$.
Regarding the proof, I'll do a sketch.
Let's consider a deviation from the above strategy.
Say that at day $t$ before the last $0 < n \leq 2^{t-1}$ machines build $Y$ instead of a copy.
In that case, you'll end up with $$ (2^{t} - n) \times 2^{d-t-1} + n = 2^{d-1} - (2^{d-t-1}-1)n < 2^{d-1} $$ copies of $Y$. By analogy we could consider 2,3,4,... deviations from this strategy and we would always get less copies of $Y$ in the end.