Question

The graphs of the equations $\begin{aligned}y&=k,\\y&=\sqrt3x+2k,\\y&=-\sqrt3x+2k,\end{aligned}$ are drawn in the coordinate plane for $k=-10,$ $-9,$ $-8,$ $\ldots,$ $9,$ $10.$ These $63$ lines cut part of the plane into equilateral triangles of side $\frac{2}{\sqrt3}.$ How many such triangles are formed?

Answer

Answer: 660

Concepts:equilateral triangle counting regions counting pairs

Difficulty rating: 2350

Small Hint:

Index one line from each family by $i,j,k$ in $[-10,10]$

Big Hint:

A smallest triangular cell occurs precisely when $i=j+k+1$ or $i=j+k-1$

Solution:

A unit triangular cell is determined by indices $i,j,k\in[-10,10]$ satisfying $i=j+k\pm1.$ For the plus sign, $-11\leq j+k\leq9.$ There are $21-|s|$ ordered pairs $(j,k)$ with sum $s,$ so this orientation contributes $\sum_{s=-11}^{9}(21-|s|)=330.$ By symmetry the other orientation also contributes $330,$ for a total of $660.$

1994 AIME Problem 6

Answer: 660

Difficulty rating: 2350

Small Hint:

Big Hint:

Solution:

Problem 6 in Other Years