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... both of these squares are quite nice.

32² = 1024, because 32=2⁵ and so we get 2¹⁰ which is 1024;

35² = 30*40+25 = 1225

That's because:

35² = (35-5)*(35+5) + 5²

That's using the difference of squares in its less common form.

2/n

So we get:

201 = 1225 - 1024 ...

Which is obviously true, but we got there with some simple mental arithmetic, and using the difference of two squares in two different ways.

Which I found satisfying.

And yes, you can use this to help calibrate your beliefs about the sorts of things I find satisfying.

3/n, n=3

@ColinTheMathmo I met a girl in freshman Philosophy class (gen ed rqt) who told me she was envious of me because I knew I wanted to be an engineer and she didn't know what she wanted to do yet. Then she told me that the number on building was the same as the class room number squared. I told her, "Don't worry, I know what you're going to be."

@ChuckMcManis I've worked with some amazing mathematicians, and the distribution of ability in mental arithmetic is strongly, *strongly* bi-modal.

It's an interesting phenomenon.

@ColinTheMathmo How would you compare their curiosity with respect to numbers? Mostly I socialize with computer programmer types, which includes a fair number of mathematicians, but the math folks are consistently quite intrigued by mathematical relationships and mapping those relationships to a variety of situations.

Do you know any mathematicians that are not curious and just grind out the applied math day after day?

@ChuckMcManis My experience is that actual research level mathematicians are not really interested in numbers at all. They are interested in structures, patterns, and relationships between them.

Even Number Theory people are mostly talking about structures such as ${\mathbb{Z}}_p$ and so forth.

For mathematicians, "doing sums" is a matter of putting those patterns and relationships into actions, and some of them just don't care enough to bother.

Of course, there's a big difference between people doing research in maths, and people who are using maths in other contexts to solve problems.

@ColinTheMathmo Agree 100% here Colin. My observation is that some people develop a fascination/curiosity about a class of 'thing' bugs, birds, numbers, chemicals, cars, or in my case systems. That curiosity drives them to spend a lot of time thinking learning about 'thing'. And often they will fall into a career that supports that curiosity. People who like numbers often become accountants, or finance people, or economists, or mathematicians.

@ChuckMcManis I think the great shame is that maths at an advanced level isn't about numbers at all. But school says "Maths is about numbers" and anyone who doesn't like numbers is pushed into something else.

It's like school and undergrad together work as a low-pass/high-pass filter combination.

People who like numbers don't often become mathematicians ... they get filtered out in undergrad.

When a mathematician is good with numbers it's usually a by-product of being good with reasoning and structure.

1/n

@ColinTheMathmo @ChuckMcManis I had a similarly-sounding epiphany that it was logic I was *truly* good at as compared to numbers. Much to the chagrin of at least one of my professors, this manifested in my switching majors to computer science XD (The joke was - and probably remains - that discrete math should be spelled "discreet" because its existence needs to be kept secret from math majors, lest they all take a sharp turn veering away from calculus once they see the FUN math lies elsewhere.)

@Zotmeister There is so much more to math than calculus, and it's annoying, enraging, and disappointing that all school math seems targeted just at that.

Art Benjamin argues that we should teach stats and probability before calculus:

https://www.ted.com/talks/arthur_benjamin_teach_statistics_before_calculus?subtitle=en

I'd argue that Logic, Graph Theory, and Number Theory are *all* potentially better choices.

But it's hard, and no matter what you choose to teach, it can be screwed up by The System(tm)

CC: @ChuckMcManis

@ColinTheMathmo @ChuckMcManis 100% agree with every word from both of you there.

I've been a proponent for a long time of logic and math being taught as *separate courses* pre-college. Any grown adult should know the opposite of "all are" is "at least one isn't", not "none are", and yet... Students should be *taught* proper negation and syllogism alongside basic arithmetic. At the other end, symbolic logic shouldn't be a 400-level uni course - it should be mastered to graduate high school!

@Zotmeister HAH !!!

Well done ... I'll edit that.

Thank you.

...

Edited ... thanks again.