We have a white table with $2$ rows and $5$ columns , and would like to colour all cells of the table according to the following rules: $\bullet$ We must colour the cell in the bottom left corner first. $\bullet$ After that, we can only colour a cell if some adjacent cell has already been coloured. (Two cells are adjacent if they share an edge.) How many different orders are there for colouring all $10$ squares (following these rules)?
Problem
Source: 2020 Dürer Math Competition Finals Day2 E12 https://artofproblemsolving.com/community/c1622639_2020_
Tags: combinatorics, Coloring