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!

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?

Problem Post 2-4: On the Escalation of P*cking Problems

Problem Posts

Image Credit: "Cardboard Box City" by James Nash. Licensed under CC BY-NC-SA 2.0.

No, I don't think they'll make you solve this problem in fifteen seconds mentally.

[Level 1]

\( \def \cm{\rm{\,cm}} \def \mtr{\rm{\,m}} \)

You may wonder why I used the P-word for something as "simple" as packing boxes. I mean, the question is straightforward... right?

THE PROBLEM
You have a big rectangular pallet upon which you must place identical rectangular boxes. You can't stack, collapse, or overlap boxes. The sides of the boxes have to parallel to one of the sides of the pallet. What's the maximum number of boxes you can place?

It's a familiar but unremarkable question, especially for many mathletes who've had to do it under time pressure. (Fifteen seconds mental? Easy!) And the real-life applications are obvious - putting things into shipping containers, for example.

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 2-3: EROs Should be Taught in High School

Problem Posts

[Level 1-2]

Disclaimer: I'm using this problem to prove a point, that is, that it's feasible, and easy to teach elementary row operations, (in particular in conjunction with Gauss-Jordan Elimination) at the high school level (at the very least for contest-involved students). While central to linear algebra, Gaussian elimination can be appreciated at a fundamental capacity. Essentially it's just a way of manipulating numbers, no different from right-to-left addition or long division.

The main point, I believe, in introducing students to GJ, is to demonstrate that any system of $n$ linear equations and $n$ unknowns can be solved (i.e. all solutions, one or infinite, found) quite uncreatively.

If you want to know how to solve the problem, the solution is at the bottom of the article.

QUESTION
(16th PMO Orals) Suppose that $w+4x+9y+16z=6$, $4w+9x+16y+25z=7$, $9w+16x+25y+36z=12$. Find $w+x+y+z$.