MathDB

Let

n \ge 2

be a positive integer. For every number from

1

n

, there is a card with this number and which is either black or white. A magician can repeatedly perform the following move: For any two tiles with different number and different colour, he can replace the card with the smaller number by one identical to the other card. For instance, when

n=5

and the initial configuration is

(1B, 2B, 3W, 4B,5B)

, the magician can choose

1B, 3W

on the first move to obtain

(3W, 2B, 3W, 4B, 5B)

and then

3W, 4B

on the second move to obtain

(4B, 2B, 3W, 4B, 5B)

. Determine in terms of

n

all possible lengths of sequences of moves from any possible initial configuration to any configuration in which no more move is possible.

Problem Statement