The Codex 3-1: How Many Shapes?

The Codex

This is the first of the articles in The Codex, which is an attempt at a comprehensive manual for intermediate to pre-Olympiad level mathletes. Accordingly, the Codex Articles will typically be produced in PDF format for easy download and printing.

This first one compiles our four-part journey through the surprisingly sophisticated art of shape counting.

Problem Post 3-1: How many Shapes, Part 4: Triangles

		Problem Posts
		Part 4 of 4 in How Many Shapes?

[Level 3]

So last time we finished off the dreary onus of counting rectangles in rectangles. Now for something completely different.

CAN WE DO BETTER?

• What if the original grid were a rectangle, instead of a square? Or an irregular figure?
• What if we were counting rectangles instead of squares?
• What if some of the grid 'wires' are missing?
• What if we're working on a triangular grid, counting triangles? Does our logic still apply?

3 Triangles!

This one was surprising to say the least. I had hoped to find a straightforward formula like in counting rectangles, but it turns out triangle counting is nastier than one might think!