2004 AMC 12A Problems/Problem 13

Problem

Let be the set of points in the coordinate plane, where each of and may be , , or . How many distinct lines pass through at least two members of ?

Solution

Solution 1

Let's count them by cases:

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• Case 1: The line is horizontal or vertical, clearly .
• Case 2: The line has slope , with through and additional ones one unit above or below those. These total .
• Case 3: The only remaining lines pass through two points, a vertex and a non-vertex point on the opposite side. Thus we have each vertex pairing up with two points on the two opposites sides, giving lines.

These add up to .

Solution 2

There are ways to pick two points, but we've clearly overcounted all of the lines which pass through three points. In fact, each line which passes through three points will have been counted times, so we have to subtract for each of these lines. Quick counting yields horizontal, vertical, and diagonal lines, so the answer is distinct lines.