An algebra problem by A Former Brilliant Member

$\begin{aligned} \text{Total number of bottles}&=2^{16}\\ \text{Number of bottles per boy}&=\dfrac{2^{16}}{2^8}\\ &=2^8\\ \text{Total number of chocolates}&=\sum_{i=1}^{2^{16}} i\\ &= \dfrac{2^{16}(2^{16}+1)}{2}\\ &=2^{15}(2^{16}+1)\\ \text{Number of chocolates per boy}&= \dfrac{2^{15}(2^{16}+1)}{2^8}\\ &=2^{7}(2^{16}+1)\\\\ \text{Average number of chocolates} \\ \text{in a bottle for a given boy}&=\dfrac{(2^{16}+1)}{2}\\\\ \text{Now group the bottles into pairs}&\text{ so that each pair is denoted as,}(i,2^{16}+1-i)\hspace{4mm}\color{#3D99F6}\small\text{where i denotes the bottle number}\\ \text{Note that the average number}&\text{ of chocolates in each pair is }\dfrac{(2^{16}+1)}{2}\\ \text{There will be } 2^{15} \text{ pairs}\text{ in total}&\\ \text{A boy can be given any } 2^7 \text{ such}&\text { pairs}\\ \text{Thus it is possible for a given boy}&\text{to have any 2 bottle numbers,}\\ \text{In fact it is possible for a given boy}&\text{to have any 128 bottle numbers of his choice}\end{aligned}$