MathDB

Suppose a table with one row and infinite columns. We call each

1\times1

square a room. Let the table be finite from left. We number the rooms from left to

\infty

. We have put in some rooms some coins (A room can have more than one coin.). We can do

2

below operations:

a)

If in

2

adjacent rooms, there are some coins, we can move one coin from the left room

2

rooms to right and delete one room from the right room.

b)

If a room whose number is

3

or more has more than

1

coin, we can move one of its coins

1

room to right and move one other coin

2

rooms to left.

i)

Prove that for any initial configuration of the coins, after a finite number of movements, we cannot do anything more.

ii)

Suppose that there is exactly one coin in each room from

1

n

. Prove that by doing the allowed operations, we cannot put any coins in the room

n+2

or the righter rooms.

Problem Statement