MathDB

On the table there are

k \ge 3

heaps of

1, 2, \dots , k

stones. In the first step, we choose any three of the heaps, merge them into a single new heap, and remove

1

stone from this new heap. Thereafter, in the

i

-th step (

i \ge 2

) we merge some three heaps containing more than

i

stones in total and remove

i

stones from the new heap. Assume that after a number of steps a single heap of

p

stones remains on the table. Show that the number

p

is a perfect square if and only if so are both

2k + 2

and

3k + 1

. Find the least

k

with this property.

Problem Statement