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 $w+x+y+z$.

Side Story 2-1: Do You Construct, or Do You Freehand?

Newton by William Blake. Public domain.

By using only compass and straightedge, you become part of an ancient tradition.

Quick Question:
Do you prefer using a compass and a straightedge to draw geometric diagrams, or would you rather freehand?

Most of the people I know freehand. I would posit that the simple reason is that geometric construction isn't a topic touched upon at great length in most of our classrooms, and dabbling with the compass is mostly done in drafting class (that is, if you had one; I didn't.). When it comes, however, to people who deal with more elaborate [Difficulty 4+ in this blog] geometry problems on a regular basis, the question does become a tad more relevant. The figures get quite complicated, and for some, the best way to make any sense out of anything is to draw them as precisely as possible. Hence constructing.

Video: Sum of Consecutive Squares

[Level 0]
Some things really are easier shown than said. Here's an uber-concise graphical 'proof' of the famous identity
\[
\sum_{i=1}^{n}i^{2}=\dfrac{n\left(n+1\right)\left(2n+1\right)}{6}
\]
(For those unfamiliar with \(\Sigma\) notation, here's a quick explanation. Trust me, it's not as scary as it looks!)