Vertical sides in hexagonal parallelogram
Source: Tuymaada 2014, Day 1, Problem 2 Juniors, Problem 4 Seniors
July 12, 2014
geometryparallelograminductioncombinatorics unsolvedcombinatoricsTuymaada
Problem Statement
A 'parallelogram' is drawn on a paper with hexagonal cells (it consists of horizontal rows of cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs. Juniors) How many vertical sides can there be in this set?
Seniors) How many ways are there to do that? [asy]
size(120);
defaultpen(linewidth(0.8));
path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle;
for(int i=0;i<=3;i=i+1)
{
for(int j=0;j<=2;j=j+1)
{
real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2;
draw(shift(shiftx,shifty)*hex);
}
}
[/asy]
(T. Doslic)