Our math prof (who moonlighted as a "mathemagician") taught us a trick for two-digit squares:<p>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.
One nice thing from knowing squares is that you can calculate for example 23 * 47 as (35 - 12) * (35 + 12) = 35^2 - 12^2.<p>But this doesn't always work and you still need to be good at adding/subtracting.
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.
73-50=23 => 23x200=4600,
50-23=27, 27^2=729
=> 73^2=4600+729=5329, all can be done in one’s head.<p>For 27^2, one can just memorize, or:
27-25=2 => 2x100=200,
25-2=23, 23^2=529.
27^2=200+529=729.<p>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?<p>There are a several little triplet "patterns" in this first batch that make it easy to this point:<p>3.1415 926 535 8 979 323 84 626 433.
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...
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.<p>the problem is that I don't know why do this.<p>nonetheless I can report that I've memorized 3 instances of two consecutive twin primes<p>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),<p>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