6 ms·

Memorizing the first 100 perfect squares (2022)

88 points by aragonite 2y ago

absolute7 2y ago
This is cool
- freedomben 2y ago
  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.
thechao 2y ago
(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).
- metacritic12 2y ago
  20ab?
  - thechao 2y ago
    Yes
quantum_state 2y ago
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 … :-).
deleted 2y ago
[deleted]
rustybolt 2y ago
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.
- amelius 2y ago
  What is wrong with computing it just as 7*23 + 40*23 ?
  - bibanez 2y ago
    Nothing, but if you know perfect squares you will do 3 additions/subtractions instead of 2 products and 1 addition
- JadeNB 2y ago
  > 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.
  - shiandow 2y ago
    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.
    - JadeNB 2y ago
      > 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.
ysofunny 2y ago
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
- deleted 2y ago
  [deleted]
- your_friend 2y ago
  reading this just reminded me how i miss talking to my friends with scientific background
  - deleted 2y ago
    [deleted]
- kstrauser 2y ago
  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.
vinay_ys 2y ago
This is beautiful. Maybe useless, but still a lot of fun to learn.
jurassicfoxy 2y ago
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.
- FredPret 2y ago
  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.
  - Arnavion 2y ago
    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 https://www.youtube.com/watch?v=5TkIe60y2GI
    - FredPret 2y ago
      Very interesting, thank you. He starts talking about this at 8:20
  - thfuran 2y ago
    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.
  - kurisufag 2y ago
    this in practice: https://github.com/ajeetdsouza/pifs https://github.com/ajeetdsouza/pifs
frankus 2y ago
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.
- JohnMakin 2y ago
  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.
  - deleted 2y ago
    [deleted]
  - bee_rider 2y ago
    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.
    - JohnMakin 2y ago
      They weren't formulae - it was literal arithmetic. Complete waste of time and space.
    - laurentlb 2y ago
      My formula for calculating a tip is `return 0`, but it's very country-dependant. (US-defaultism?)
      - spookie 2y ago
lupire 2y ago
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)
apalmer 2y ago
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...
dxbydt 2y ago
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.
- madcaptenor 2y ago
  When I did math competitions I remember getting the advice to know the prime factorization of the current year.
  - dxbydt 2y ago
    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.

88 points by aragonite 2y ago

absolute7 2y ago

This is cool

freedomben 2y ago
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.

thechao 2y ago

(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).

metacritic12 2y ago
20ab?
- thechao 2y ago
  Yes

quantum_state 2y ago

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 … :-).

deleted 2y ago

[deleted]

rustybolt 2y ago

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.

amelius 2y ago
What is wrong with computing it just as 7*23 + 40*23 ?
- bibanez 2y ago
  Nothing, but if you know perfect squares you will do 3 additions/subtractions instead of 2 products and 1 addition
JadeNB 2y ago
> 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.
- shiandow 2y ago
  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.
  - JadeNB 2y ago
    > 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.

ysofunny 2y ago

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

deleted 2y ago
[deleted]
your_friend 2y ago
reading this just reminded me how i miss talking to my friends with scientific background
- deleted 2y ago
  [deleted]
kstrauser 2y ago
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.

vinay_ys 2y ago

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

jurassicfoxy 2y ago

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.

FredPret 2y ago
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.
- Arnavion 2y ago
  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 https://www.youtube.com/watch?v=5TkIe60y2GI
  - FredPret 2y ago
    Very interesting, thank you. He starts talking about this at 8:20
- thfuran 2y ago
  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.
- kurisufag 2y ago
  this in practice: https://github.com/ajeetdsouza/pifs https://github.com/ajeetdsouza/pifs

frankus 2y ago

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.

JohnMakin 2y ago
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.
- deleted 2y ago
  [deleted]
- bee_rider 2y ago
  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.
  - JohnMakin 2y ago
    They weren't formulae - it was literal arithmetic. Complete waste of time and space.
  - laurentlb 2y ago
    My formula for calculating a tip is `return 0`, but it's very country-dependant. (US-defaultism?)
    - spookie 2y ago

lupire 2y ago

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)

apalmer 2y ago

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...

dxbydt 2y ago

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.

madcaptenor 2y ago
When I did math competitions I remember getting the advice to know the prime factorization of the current year.
- dxbydt 2y ago
  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.