Painting balls into 4 colours
Source: 2013 China TST Quiz 1 Day 1 P3
March 30, 2013
inductioncombinatorics proposedcombinatorics
Problem Statement
There are balls numbered , respectively. They are painted with colours, red, yellow, blue, and green, according to the following rules:
First, randomly line them on a circle.
Then let any three clockwise consecutive balls numbered , in order.
1) If , then the ball is painted in red;
2) If , then the ball is painted in yellow;
3) If , then the ball is painted in blue;
4) If , then the ball is painted in green.
And now each permutation of the balls determine a painting method.
We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods.Find out the number of all distinct painting methods.