Question 44
Let $k$ be a positive integer. Determine the least integer $n \ge k + 1$ for which the game below can be played indefinitely:
Consider $n$ boxes, labelled $b_1, b_2, \dots, b_n$. For each index $i$, box $b_i$ contains initially exactly $i$ coins. At each step, the following three substeps are performed in order:
(1) Choose $k+1$ boxes;
(2) Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_i$, a number of $i$ coins.
(3) If one of the boxes is left empty, the game ends; otherwise, go to the next step.
Consider $n$ boxes, labelled $b_1, b_2, \dots, b_n$. For each index $i$, box $b_i$ contains initially exactly $i$ coins. At each step, the following three substeps are performed in order:
(1) Choose $k+1$ boxes;
(2) Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_i$, a number of $i$ coins.
(3) If one of the boxes is left empty, the game ends; otherwise, go to the next step.