# Grid $2\times3$ colored with four colours

I have a $2\times3$ grid which is colored with four colours. How many colourings do there exists such that every two adjacent squares have different colours? I don't know how to start.

• Start by trying to solve similar problems for smaller grids and fewer colors. – MJD Jan 14 '17 at 18:39
• @MJD What is the basic idea? – Problemsolving Jan 14 '17 at 18:45
• This idea should work but not in general: Top middle cell has 4 choices, neighbors have 3 choices, then take cases for the last two cells. – greenturtle3141 Jan 14 '17 at 18:47

You have $4\cdot 3=12$ ways to color the $1$st column. Now given that you've made your choice of colors for the $1$st column, the number of ways to color the $2$nd column will drop down from $12$ to $7$. The same happens for the 3rd column, once you've colored the $2$nd one . So the total number is $$12\cdot 7 \cdot 7=588.$$
• Let A, B,C, D be the four colors. Suppose the $1$st column is (A,B) (A on the top, and B on the bottom). Can you list the possibilities for the $2$nd column? – MathChat Jan 14 '17 at 20:23