1994 AIME Problem 6

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

Index one line from each family by

i,j,k

in

[-10,10]

A smallest triangular cell occurs precisely when

i=j+k+1

or

i=j+k-1

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.

Problem 6 in other years