MathDB

Let

k

be a positive integer. Determine the least positive integer

n

, with

n\geq k+1

, for which the game below can be played indefinitely:
Consider

n

boxes, labelled

b_1,b_2,...,b_n

. For each index

i

, box

b_i

contains 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.
Proposed by Demetres Christofides, Cyprus

Problem Statement