Banners cover grid table
Source: CSMO 2019 Grade 10 Problem 4
July 31, 2019
combinatorial geometrycombinatorics
Problem Statement
As the figure is shown, place a grid table in horizontal or vertical direction, and then remove arbitrary one square on its four corners. The eight different shapes consisting of the remaining nine small squares are called banners.
[asy]
defaultpen(linewidth(0.4)+fontsize(10));size(50);
pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9);
draw(B--C--H--J--N^^B--I^^D--N^^E--M^^F--L^^G--K);
draw(Aa--Ca--Ha--Ja--Aa^^Ba--Ia^^Da--Na^^Ea--Ma^^Fa--La^^Ga--Ka);
[/asy]
[asy]
defaultpen(linewidth(0.4)+fontsize(10));size(50);
pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9);
draw(B--Ca--Ea--M--N^^B--O^^C--E^^Aa--Ma^^Ba--Oa^^Da--N);
draw(L--Fa--Ha--J--L^^Ga--K^^P--I^^F--H^^Ja--La^^Pa--Ia);
[/asy]
Here is a fixed grid table. Find the number of ways to cover the grid table completely with 18 banners.