Memorizing the first 100 perfect squares (2022) | Hacker News Reader

Memorizing the first 100 perfect squares (2022)

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88 pointsaragonite55 comments

Our math prof (who moonlighted as a "mathemagician") taught us a trick for two-digit squares:

For e.g. 23 x 23, subtract three from the first number, add three to the second, and then add 3² to the product. So 20 x 26 + 9, the idea being that multiplying by a multiple of 10 is easier to do mentally.

This would have made my college career significantly easier. We were limited on scratch paper (dumb policy) on many tests, and I have to write out arithmetic like this long-hand style, which took up a ton of space.

I think these sort of mental math tricks are mostly useful when you have a case where you’ve got a formula that you often have to run, and it is similar enough every time that you can pre-do the algebra.

The one everyone knows is calculating a tip (bump the decimal place left once and double the result, round depending on how nice you feel). Might be applied multiple times per day, depending on your dining habits.

I bet the formulae you had to use on your tests were worth doing by hand.

They weren't formulae - it was literal arithmetic. Complete waste of time and space.

My formula for calculating a tip is `return 0`, but it's very country-dependant.

(US-defaultism?)

You may tip anywhere you find the service pleasant.

(EU guy who has worked as a waiter)

I recently leafed through "Mathematics for Engineers"[0], originally from 1926, which begins with many such methods for making arithmetic easier. Though maybe not always more space efficient.

Luckily I got to use a calculator for school.

[0]https://archive.org/details/dli.ernet.11725/page/n9/mode/1up

If you look at the way they teach kids math on paper today, they are just teaching them how to do math in your head. The idea being that if your hands were free you’d use the calculator on your phone.

When I first encountered the outrage over New Math my first thought was that this is how I avoid embarrassing myself in checkout lines. Do I have enough cash to pay for this stuff in my hands?

I do the less efficient quadratic form. It works but is slower.

20² + 20 x 3 x 2 + 3²

Neat! This is (a+b)(a-b) + b² = a²-b² + b² = a² and in your eg, a=23 and b=3

In addition to using this trick for getting to multiples of 10, I used it to compute the product of two numbers by leveraging their proximity to a number in between whose square I knew. For eg if I need to multiply 23 by 27, I instead see it as (25-2)(25+2), which is 25²-2² = 621.

(using that same trick from the article to calculate squares of numbers that end in 5)

For ref: https://en.m.wikipedia.org/wiki/Difference_of_two_squares#Me...

That’s a neat one!

(a + b)(a - b) = a^2 - b^2

a^2 = (a + b)(a - b) + b^2

So pick a value of b that makes a - b end in a zero.

FWIW you can also pick a value of b to make a + b end in a zero, which keeps b at 5 or less.

One nice thing from knowing squares is that you can calculate for example 23 * 47 as (35 - 12) * (35 + 12) = 35^2 - 12^2.

But this doesn't always work and you still need to be good at adding/subtracting.

> But this doesn't always work and you still need to be good at adding/subtracting.

It does always work. If you mean that not every integer product can be written this way, you're right; 23 × 46 can't, unless you're willing to memorise squares of half-integers. But, if avoiding non-squaring multiplication is really key, then you can still just write 23 × 46 = 23 × 47 - 23, and then compute 23 × 47 = 35^2 - 15^2 as you suggest.

Or you could do ((46+23)^2 -(46-23)^2) / 4

Or one of several similar formulae, but each has its own pros and cons.

> Or you could do ((46+23)^2 -(46-23)^2) / 4

Sure, of course that works algebraically, though this method will always involve at least one bigger square, and division by 4, which, if working in base 10, can be implemented with exactly the computational complexity of multiplying by 25—so perhaps is also meant to be avoided, if we're trying to avoid multiplication! As you say, there are pros and cons of all approaches, including just doing the multiplication.

I mean the initial method is just ((46+23)/2)^2 - ((46-23)/2)^2 so you kind of need to be good at halving to use that method in the first place.

What is wrong with computing it just as 7*23 + 40*23 ?

Nothing, but if you know perfect squares you will do 3 additions/subtractions instead of 2 products and 1 addition

I'm good at arithmetic, to the point that my buddy called from across the country a few days ago to have me perform "hard" math questions for his kids. It's always a relief when they pick random numbers like "what's 83 times 97?" Whew. Well, 90**2 - 7**2 = 8051. As a bonus, when you explain that method, it sounds even more impressive to them: "you mean, you just know what 63 squared is?" Sure, but it didn't happen overnight.

I think your star star just rendered as a star.

Oops, good catch.

That close to 100, I'll just do 83*100 - 83*3

That takes me longer for whatever reason. It would also require me to identify that as a special "close to 100" case I'd have already calculated it the other way by the time decided to swap in that algorithm. Basically, branching takes me too many cycles.

Hmm, this looks like a neat trick to make hardware multiplication faster. Say, for 8-bit numbers it would require 3 8-bit additions/subtractions and one 16-bit addition, one 9-bit shift, and two lookups in a 512-byte table of squares, and zero conditional processing, as opposed to 8 16-bit additione, 8 16-bit shifts, and 8 LSB tests that a naive iterative algorithm would do.

I wonder what real integer multiplication hardware uses.

It all depends on how you can parallellize it.

(10a + b)^2 = 100a^2 + b^2 + 2ab

So... 73^2 is 4900 + 9 + 420 = 5329. The really nice part is getting estimates for square roots of numbers.

So, sqrt(3895)? 60^2 + 120n = 3600 + 120n => n=2; that's 3844 (from above); the difference is 51; the residual estimate is then: 62 51/(62*2).

20ab?

Yes

I give amc & mathcount mocks to kids every weekend & time them. There was one question everyone got under 5 seconds. I was like how did you guys do it so fast. Apparently aops had told them every year will have a special property that'll forsure be on the test. So that's why they all knew 45^2 was 2025.

When I did math competitions I remember getting the advice to know the prime factorization of the current year.

45^2 = 2025 => must be the sum of first 45 odds => 1 + 3 + ... + 87+ 89 = 2025

while its not pythagorean, 40^2 + 20^2 + 5^2 = 2025.

877 is another bullcrap that has shown up on a bunch of these tests. 877 is a prime, and four times 877 is 3508 = sum of all the divisors of 2025.

there's a bunch more i shared with them.

73-50=23 => 23x200=4600, 50-23=27, 27^2=729 => 73^2=4600+729=5329, all can be done in one’s head.

For 27^2, one can just memorize, or: 27-25=2 => 2x100=200, 25-2=23, 23^2=529. 27^2=200+529=729.

As long as one knows square of numbers up to 25, it is done for all up to 100 and more … :-).

I previously memorized pi to 100 decimals. Was fun, and now I'll never forget 3.1415926535897932384626433 ... how many is that? 25?

There are a several little triplet "patterns" in this first batch that make it easy to this point:

3.1415 926 535 8 979 323 84 626 433.

I'm guessing the answer to this question is no, but... for an irrational number like pi, can we guarantee that a certain digit sequence will occur somewhere in its infinite reaches?

That would be a wildly impractical but very fun way to encode information - if everybody had a couple of petabytes of pi on their harddrive one day, you could just send the starting and ending digit to communicate an arbitrary amount of information this way.

Of course you'd first have to search through the whole universe of digits to find a sequence that's just right.

Such numbers are called "normal numbers". Pi likely is one based on all the digits we've computed for it, but we don't have any way to prove a given number like pi is normal or not.

Relevant Numberphile video: https://www.youtube.com/watch?v=5TkIe60y2GI

Very interesting, thank you. He starts talking about this at 8:20

1.01001000100001... is infinite and non-repeating, but doesn't contain all substrings. A number that does is called a "normal" number, and it's not known whether pi is. It seems to be pretty normal though.

this in practice: https://github.com/ajeetdsouza/pifs

That reminds me of how on The IT Crowd, the new -- exceedingly long -- Emergency Services number replacing 999 was fairly easy for Moss to remember. Being divided into groups of at most 5 digits probably helped.

No need to memorize

(10a+5)² = [a * (a+1)][25]

(a+1)² = a² + a + (a+1)

(a-1)² = a² - a - (a-1)

(a+2)² = a² + 4(a+1)

(a-2)² = a² - 4(a-1)

I am not knocking the article but seems like if you are going to dedicate the effort of learning quick mental math it's probably more efficient to 'just' know how to multiple 2 digits quickly than specifically focusing on 2 digit squares...

This is beautiful. Maybe useless, but still a lot of fun to learn.

This is cool

Why is it cool? What is it about TFA that interests you? What does it do that other resources have not done? Does it help you think about the problem in a different way?

I've always been fascinated with the perfect squares, and various patterns that arise from observing them. I love how the article examines the patterns and then extrapolates the study of the patterns to a tool for memorization.

I memorized all primes up to 127. and with a bit of effort I may come up with all the primes up to 307 (then 311,313, iirc) but I need pen and paper to double check.

the problem is that I don't know why do this.

nonetheless I can report that I've memorized 3 instances of two consecutive twin primes

11,13,17,19, then 101,103,107,109 (which just raises questions that I can only aspire to ask, nevermined answering, about the what, why, and how of decimal system),

and then 191,193,197,199. the next prime is 211. but the cool thing is how 210 = 2*3*5*7, which are all primes before the first double twin prime

reading this just reminded me how i miss talking to my friends with scientific background

I know exactly why I do this. I use to deliver pizza. My store was on the edge of a medium-sized city, and a lot of our customers were quite the drive away. I'd get incredibly bored on these long drives. Somewhere along the way, I'd start calculating what percent of my shift I'd worked that minute, and I'd try to get it to 3 decimal places before the next minute came. Again, no reason whatsoever, just bored. And then at some point I thought it'd be more interesting to calculate (number of minutes worked) / (number of minutes remaining), which is fun because the ratio grows really slowly at first and then very quickly. I realized the first step in that was dividing out the common factors, so 248 minutes / 232 minutes was 31 / 29. (Yes, I'd been taught that in class, but it's one thing to have a teacher make you do something and another to realize why you'd ever want to do it voluntarily.)

Do that long enough and you find patterns, like 7 * 11 * 13 = 1001. If you ever end up calculating n / 11, it's approximately the same as n * 7 * 13 / 1000. E.g., 3 / 11 ~= .273. Or take 27 * 37 = 999. Now n / 37 ~= n * 27 and shuffle the decimals. 7 / 37 ~= 7 * 27 / 1000.

And that's how I ended up reasonably good at mental arithmetic, and memorizing a frankly unnecessary number of squares, and being able to factor lots of numbers at a glance (or recognize that they're prime). I was awfully bored for an awfully long time.

Boredom can be great.

Imagine being an ancient Babylonian or Greek with nothing to do but wait for your grapes to grow. No wonder they came up with lots of great stuff.

> Boredom can be great.

I agree, and I worry that the absence of boredom-time, especially for kids and adolescents, will turn out to be a bad thing.

I've never met a kid or adolescent who shares this concern. :)

191, 193, 197, 199 are also twin twin prime pairs

These primes have to be 11,13,17,19 + 30k, due to division by 3.

5 is half of 10, so we easily rule out exactly the number 10x+5.

49 = 7²,

77 is a multiple of 7, of course.

98 = 2 • 7²

In order to avoid a multiple of 11 interfering, we need the gap preceding 11•10

99 is (10+1)(10-1) = 10²-1 and 100 is 10²

, so that clears out space for primes after 100.

Also, You might enjoy "Paterson Primes" https://m.youtube.com/watch?v=jhObLT1Lrfo

I'm intersted in why (and how) you reason that thing about "multiple 11s interferring"

so I'll tell you how I memorized the primes between 2 and 127.

to begin: all primes less than ten: 2,3,5,7. I seem to have learned these by rote memorization. but then, the 'fun coincidences' begin.

because 5 is a multiple of "ten", all primes after 10 can only end in 1,3,7,9. this is a key somehow.

this gives the first fun coincidence: that 11 and 101 are both prime.

after 19 comes 23. similarly (and this is the part where every self-respecting numberphiliac waves their hands a little, in excitement I hope), 113 ("one one three") is the next prime after "one oh nine".

notice that 23,29,31,37 are as much prime as 30+{23,29,31,37} = 53,59,61,67

this covers almost all of them. but we're missing primes in the 40s, and the 70s. as well as primes in the 80s and 90s (only 3 primes: 83, 89, and 97. I have no tricks to remember these 3. only rote repetition)

the forties and seventies, are a very compact: both 1,3. but only 47, and only 79; because, well, what fun! seven squared and seven eleven are just so simple, like low-hanging fruit in the garden of prime-number coincidences

I confess that none of those patterns resonate with me at all. I’m not sure if I can even see that patterns in them that you see.

Conversely, to me, 9733 “looks prime”. I couldn’t explain that if I had to. It just does.

I like that there’s enough prime real estate for us all to see it differently.

I've realized that the way I explained the "patterns" is very different from what I was doing when I noticed them. which makes sense as I was not trying to explain that little "game". it's not a game it's just counting and writing primes in some base, and a dash instead for every composite. as the gaps get bigger I draw a triangle instead of 3 dashes; and then any glyph with as many strokes as dashes.

doing it in different bases helps to think about the numbers without their decimal representation. which I guess is the point, and might be helping me imagine patterns that may or may not be there. if I learned anything from doing this is that there is no pattern, it just seems as if there is one but it's never really there.

I like to think that the "patterns" expire, they have only so many uses in them. often just one use which breaks the point of being a 'pattern', but that's primes; is all am saying.

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