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!

Problem Post 2-6: How Many Shapes, Part 3: Rectangles in Not-Rectangles

		Problem Posts
		Part 3 of 4 of How Many Shapes?

[Level 2-3]

While it would be far more fun for me if I jumped from topic to topic, I believe I owe it to you, dear readers, to flog this horse until it's brain dead. Completeness is a virtue.

So last time we finished off the case where we count rectangles in rectangles, even when some 'matchsticks' or grid wires are missing. Now onwards to rectangles in non-rectangles.

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?

2 Rectangles in Non-Rectangles

This is ad hoc land. The base technique would again be to count one-by-one, but as we will see there are tricks for certain special irregular figures.

Problem Post 2-5: How many Shapes, Part 2: Rectangles in Rectangles

		Problem Posts
		Part 2 of 4 of How Many Shapes?

[Level 2-3]

So in my previous Miscellaneous post, we solved a fairly simple Internet puzzle, counting the number of squares in a square grid. But now, we ask ourselves if we can do more:

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?

Miscellaneous 2-1: How many Shapes, Part 1: How many Squares?

Part 1 of 4 of How Many Shapes?

[Level 0]

I can't really classify this a Problem Post, because its roots lie more in Internet culture than in math contests.

Remember this nasty little thing making the rounds on social media?

Once you read this post, 100% will put this puzzle behind us.

We could just manually count squares, but that's boring, and we want to find a method to do this for huge squares.

PROBLEM 1
Devise a method to rapidly solve problems of this type, no matter the size of the square grid.