Forty-five Squared

December 30, 2024

1. "Remember twenty-four?"

2. Competition Math

3. Not a fin bro, nor am I a fish

4. "Just trust me, you'll be fine"

"Remember twenty-four?"

When I was very little, probably in early elementary school, my dad would teach me math at home. I remember being rather ambivalent about it then, but those teachings would have a much longer-lasting impact than little-me could have predicted. It definitely put me on the path for competition math later on.

One of the fun math tricks he taught me was squaring integers that ended in 5. Take 45², for example. First, it must end in 25. Then, we take whatever precedes 5 in the integer—in this case, we have 4. Take that and multiply it by the successive integer, so in this case, we have 4⋅(4+1) = 20. Append that to the 25, and we get 45² = 2025!

Another example with 65²: Ends in 25, then we take whatever precedes the 5 in the integer—so this time we have 6—multiply it by the successive integer, so 6⋅(6+1) = 42. Append the 42 to the 25 from before and we get 65² = 4225!

I was curious about whether this trick would generalize to larger values, which was what had prompted me to try 165, and then 1655. Turns out that the trick still works!

So if we have any integer that ends in 5, we can write it as a5, such that “a” represents the portion that precedes the 5. We can also write a5 = 10a+5. Squaring both sides and expanding, we get:

This result is important, because a² + a can be written as a(a+1), which represents the procedure in taking whatever precedes 5 and multiplying it by the successive integer. And then, what the 100 does is it shifts the result of a(a+1) to the left by two, which makes room for the tens and ones digit. Finally, the 25 is appended and fills up the tens and ones digit. Yippe!

The upcoming year is a square year, which is pretty cool! Other fun things to notice are:

Generally, I like to notice fun numerical things, like in phone numbers, serial numbers, or codes. Another favorite is in the game 24 with a standard set of playing cards, which is another numbers-related lesson my dad taught me. Once on a road trip, I forgot to bring playing cards, so I instead played 24 using numbers on the license plates of passing cars. The numbers were random enough, although if there were zeroes or too many letters, that was another surprise element.

24 was perhaps my official introduction to playing with numbers. I remember later on, I was playing around with powers of 11 when I noticed that the first few powers had formed Pascal’s Triangle! Just writing them out in a triangle and noticing the numbers fall beautifully was amazing.

I excitedly told a friend, thinking that I had discovered something novel. But then they said something along the lines of, "What, you didn’t know that already?" I don't think they intended anything malicious with that statement, but I remember feeling a little inferior. The spark of having "discovered" that trick went away.

I didn't realize it then, but in retrospect, I think that comment killed my confidence a lot. I don't remember having felt much joy learning math after that—it was just grinding chapter after chapter in the AoPS series, taking test after test with the plethora of AMC and mock exams available. Even though the concepts hidden in each problem were objectively super cool, the spark was gone.

At least for now.

Competition Math

The competition math community, at least here, can be extremely competitive. While part of that is inherent from there being a competition in the literal sense, there is a certain demographic where it is, quite unfortunately, quite common to compare kids’ scores. I'll hear people calling each other stupid just because they "sillied" or got one less question correct than someone else. I'll hear gossip about how someone had a surpringly good performance, or celebrations when another had "underperformed" in a competition.

One year, I was compared to this guy, because we were known to be neck-and-neck with each other. I got second place, which was a little unfortunate, but fine. But the parents were discussing, and it's quite common to almost... worship? Or rather, praise the parent of the kid who won. The parent of this guy proceeds to say something along the lines of, "Second place is extremely impressive for a girl!" Which uh, yikes!

That incident was when I became a lot more aware of the lack of girls in the STEM field. Later on, my friend had quit competition math, and it became a little lonely at times. With my worth tied to a score, the very philosophy I would turn against later, it kept me going—climbing higher and higher with no end in sight.

While I do see merit in comparisons, I think it could also be demotivating if used poorly. Comparisons are meant to inspire, not push down on others.

At some point, I was going to write about how we should all be like the Jack-in-the-box mascot, such that his round head represents well-roundedness, while his cone-shaped nose/hat represents one's "spike" or "main." But in the end, I don't think it matters very much. I think it's a lot more important to find things that we enjoy doing. Rather than strive to be well-rounded or have a spike, I believe it's more helpful to just do, and doing things happen to make us well-rounded or build a spike.

It's a lot more important to take action than just have empty words. For me, this realization came through participating in years of competition math. Competition math was a constant in my life, though my perspective towards it has drastically changed. Rather than a source of conflict, it is a place to connect.

Not a fin bro, nor am I a fish

In freshman year, I took Financial Literacy, and while there were important lessons about how to deal with finance, it's the life lessons that had a much larger impact on me. One day, the teacher instructed us to hold our palms out and close our eyes. When he said "up," we would point our palms up, and when he said "down," we would point our palms down. Simple enough.

When we began, I couldn't really see the point. I thought it was quite a funny game, because it was similar to Simon Says. I tried counting the rhythm at which he was saying "up" and "down," deciphering whether there was a pattern or not. After a few minutes and failing to find a discernible pattern, I wondered what the point was. Was it to test our patience, to see who would last the longest? I must've smiled then—the ones who were still going would not know how much longer they had to go. I found it entertaining to try figuring out what kind of social experiment this was.

He told us to open our eyes and began pointing out the people he would hire to his company if he had one. For one of my classmates, they were chosen because they held their hands out firmly and diligently followed his directions the whole time rather than gradually flopping his arms like a fish. To my surprise, he also pointed me out, because I would be a fun person. He mentioned how I found the fun even in the most boring of tasks. When I was puzzling out the purpose of this activity, I must have shown it on my face. That's when I realized the purpose of this experiment was to demonstrate just how much you can tell about someone based off the simplest of interactions.

And it's not really something I thought about much before, but it is a characteristic I'm proud of. Since it's relevant, in hindsight, it's probably part of what kept me going in competition math. Even though there were so many times I wanted to quit, I kept going because there was still something in me that saw the spark.

"Just trust me, you'll be fine"

There’s a lot of little things in life that have built me up to the way I am now.

When I started this blog, it was actually in the hopes of learning how to write better. But of course, it became so much more than that—it was a timeline of change, full of little lessons I picked up over the year. It was a symbol of progress as I added more elements to the blog. It was a place where I forced myself to have coherent thoughts, because I was not going to let myself release word vomit.

It had become a part of me.

In July of 2022, I started my online journal, which is similar to my posts here but a lot more disheveled. Before then, I had multiple physical journals that have barely any reasonable content I can grasp. Not that the online journal really did either, but I wrote in it so much that Google Docs literally stopped me from writing any more.

Quite literally, this is the end of beginning. I think both writing in that journal and here kept me sane. Reading things from a year ago is a little sad, but it's also quite amazing how far we can come in a year. I have no idea what is going to happen in the upcoming year, but based on the many things I could tell myself a year ago, it'll all be okay.

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