Squiggle is composed of six equilateral triangles
Source: Baltic Way 2007
November 30, 2010
geometrygeometric transformationrotationreflectionparallelogrammodular arithmeticcombinatorics proposed
Problem Statement
A squiggle is composed of six equilateral triangles with side length as shown in the figure below. Determine all possible integers such that an equilateral triangle with side length can be fully covered with squiggles (rotations and reflections of squiggles are allowed, overlappings are not).[asy]
import graph; size(100); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black;
draw((0,0)--(0.5,1),linewidth(2pt)); draw((0.5,1)--(1,0),linewidth(2pt)); draw((0,0)--(3,0),linewidth(2pt)); draw((1.5,1)--(2,0),linewidth(2pt)); draw((2,0)--(2.5,1),linewidth(2pt)); draw((0.5,1)--(2.5,1),linewidth(2pt)); draw((1,0)--(2,2),linewidth(2pt)); draw((2,2)--(3,0),linewidth(2pt));
dot((0,0),ds); dot((1,0),ds); dot((0.5,1),ds); dot((2,0),ds); dot((1.5,1),ds); dot((3,0),ds); dot((2.5,1),ds); dot((2,2),ds); clip((-4.28,-10.96)--(-4.28,6.28)--(16.2,6.28)--(16.2,-10.96)--cycle);[/asy]