MathDB

There is a deck of cards placed at every points

A_1, A_2, \ldots , A_n

and

O

, where

n \geq 3

. We can do one of the following two operations at each step:

1)

If there are more than 2 cards at some points

A_i

, we can withdraw three cards from that deck and place one each at

A_{i-1}, A_{i+1}

and

O

. (Here

A_0=A_n

and

A_{n+1}=A_1

);

2)

If there are more than or equal to

n

cards at point

O

, we can withdraw

n

cards from that deck and place one each at

A_1, A_2, \ldots , A_n

. Show that if the total number of cards is more than or equal to

n^2+3n+1

, we can make the number of cards at every points more than or equal to

n+1

after finitely many steps.

Problem Statement