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.

Problem Post 2-1: Factor Sums, and the Distributive Law

Problem Posts

[Difficulty 1; some parts Difficulty 4]

Who knew power series multiplication could help you at the grocer? (see Question 2)

Just a quick post for younger readers. Often one will find oneself using `brute force' approaches when obvious tricks and shortcuts exist. Most of the time, this is justified -- many tricks are usually too arcane to remember, or too impracticable to execute realistically. This is neither. It's fast, simple, and it could save you in a Do-Or-Die (I've used it before; our team won!)

What is a factor?

First things first, right? Not everybody defines the word “factor” the same way. I will be using the convention used in most local contests: A factor of an integer $n$ is a positive integer $a$ for which there exists an integer $b$  such that $n=ab$. So by our convention $4$ and $-4$ both have exactly three factors: 1, 2, and 4. We do not consider $-2$ as a factor, even if it divides both $4$ and $-4$. On the other hand, a factor of a number is a proper factor iff it is not the number itself. Often 1 is also not considered a proper factor. With this ambiguity, however, the term is not used too often in contests.

Problem 1-7: Back-Engineering

Problem Posts

[Difficulty 2] [PDF]

QUESTION
(Adapted from UVA 10784) The number of diagonals of an $n$-gon is not less than $N$. Find a closed-form expression for the minimum possible value of $n$.

Laconic Solution Sketch
A polygon has an integral side number. Use bounds.