Toothpick Triangles
Source:
January 3, 2009
algebrapolynomial
Problem Statement
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have rows of small congruent equilateral triangles, with small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of small equilateral triangles?
[asy]unitsize(15mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));pair Ap=(0,0), Bp=(1,0), Cp=(2,0), Dp=(3,0), Gp=dir(60);
pair Fp=shift(Gp)*Bp, Ep=shift(Gp)*Cp;
pair Hp=shift(Gp)*Gp, Ip=shift(Gp)*Fp;
pair Jp=shift(Gp)*Hp;
pair[] points={Ap,Bp,Cp,Dp,Ep,Fp,Gp,Hp,Ip,Jp};draw(Ap--Dp--Jp--cycle);
draw(Gp--Bp--Ip--Hp--Cp--Ep--cycle);for(pair p : points)
{
fill(circle(p, 0.07),white);
}pair[] Cn=new pair[5];
Cn[0]=centroid(Ap,Bp,Gp);
Cn[1]=centroid(Gp,Bp,Fp);
Cn[2]=centroid(Bp,Fp,Cp);
Cn[3]=centroid(Cp,Fp,Ep);
Cn[4]=centroid(Cp,Ep,Dp);
label("",Cn[0]);
label("",Cn[1]);
label("",Cn[2]);
label("",Cn[3]);
label("",Cn[4]);
for (pair p : Cn)
{
draw(circle(p,0.1));
}[/asy]