Why visualisation is soooo powerful

(This article was written on 1 Jan 2025 before it was migrated here) 

Today I'm gonna talk about a super-simple but super-overpowered thing called visualisation.


Let's begin with the following problem.

(22SLC1): A $\pm 1$-sequence  is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ such that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and 
$$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Yeesh, what a statement. We have a very limited yet free sequence, and we want to choose elements in it (also under a restriction) to maximise a given quantity (which happens to look somewhat terrifying).
How do we even begin to approach this?

Well, we can try to visualise it. In fact, consider this alternative version in the same shortlist (This was the version given to USA MOP students):


$2022$ buckets of water are arranged in a row, each coloured either red or blue. Sally the salmon plays a game in the following way: first, she chooses any bucket she likes to start in. Then, any number of times she may jump either to the next bucket in the row, or across it to land in the bucket after that. (She may not jump across more than one bucket.) At any point, she may finish the game. At that time, her score is the absolute value of the difference between the number of red and blue buckets she visited during the game. Determine the largest $C$ so that no matter how the buckets are coloured, Sally can achieve a score of at least $C$.

And all of a sudden, the problem looks so much more approachable.
In fact, oftentimes, the crux of a problem is simply finding a good way to visualise the given conditions.

I wont overshadow the visualisation idea with the bulk of the solution (although the way I did it was quite instructive I feel, despite my writeup being super long - I might add it as a part 2 to this post, if it is requested). So for now, here's an 

Exercise: Try to prove the problem in the second formulation.

In other news, Happy 2025! It's gonna be nice to see what this year gives to us, and may we all truly xook like skibidi in 2025.

I'll give you guys a parting gift- my first ever solve from 2025 (also involving visualisations!)

Problem(2024 Philippines/1): Let $\mathbb{Z}^2 \to \mathbb{Z}$ be a function satisfying
$$f(x+1, y) + f(x, y+1) + 1 = f(x, y) + f(x+1,y+1)$$ for all integers $x$ and $y$. Can it happen that $|f(x, y)| \le 2024$ for all $x, y \in \mathbb{Z}$?
On that note, bye!!




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