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What are some interesting mental math techniques that you know?

Here's one that I got from my Grandmother who got it from a book: To square a two-digit number (from $26$ to $49$), take the number minus $25$ and put that in the first two digits, and then add the square of $50$ minus the number: $$(\text{number}-25)\times100+(50-\text{number})^2$$ For example, to do $47^2$ we have $47-25=22$ for the first two digits and $(50-47)^2=9$ for the last two so we get $47^2=2209$.

Bonus points if you include justification! For this trick, $$100(n-25)+(50-n)^2=100n-2500+2500-100n+n^2=n^2$$

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    $\begingroup$ Why the downvotes? $\endgroup$ – user142299 Apr 18 '14 at 0:53
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    $\begingroup$ this question is opinion based, which is against the guidelines of the math stack exchange. $\endgroup$ – Sidd Singal Apr 18 '14 at 1:19
  • $\begingroup$ @mathguy There's nothing opinionated about a mental math trick. It either works or it doesn't. How could this possibly result in arguments? $\endgroup$ – user142299 Apr 18 '14 at 2:18
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    $\begingroup$ There it's not opinion based anymore. Please don't close. $\endgroup$ – user142299 Apr 18 '14 at 2:22
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    $\begingroup$ If you look at the "Related" questions listed on this page, you'll see that there have been many questions on mental math on this site already. It might be worth having a look at a few of them, to see that we are not creating duplications --- indeed, it is possible that the current question should be closed as a duplicate. $\endgroup$ – Gerry Myerson Apr 20 '14 at 14:01
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Of course, we have the classic trick $$9\times n=(10\times n)-n$$ which works because of distributivity. This can be generalized as follows: $$99n=100n-n$$ so for example $$99\cdot 54=5400-54=5346$$ It also helps simplify generic calculations. E.g., $$17\cdot 8=17\cdot 10-17\cdot 2=170-34=136$$

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enter image description here

Makes multiplication of multi-digit numbers easier. The above is the following problem:

   14759
x 365421

This is how they teach multiplication in Japan. You may be thinking, you draw this in your mind? No, there's a shortcut for this method.

Take for instance:

  21
x 32

You can draw it to get the answer. But the drawing is basically giving you a simpler way of solving it. This is how you solve it:

http://sketchtoy.com/60373071

Here's a little tougher one that I did mentally:

http://sketchtoy.com/60373153

Makes mental multiplication of multi-digit numbers easier.

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  • $\begingroup$ You can visualize a $5\times 6$ array of numbers in your head?! $\endgroup$ – user7530 Apr 18 '14 at 0:38
  • $\begingroup$ @user7530 Obviously not this big but for 3/4-digit numbers it's useful. $\endgroup$ – Shahar Apr 18 '14 at 0:39
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    $\begingroup$ So... the standard multiplication algorithm, but less efficient. Looks pretty, though, and probably useful as an educational tool more than anything. $\endgroup$ – Omnomnomnom Apr 18 '14 at 0:41
  • $\begingroup$ @Omnomnomnom If I did 721 x 341 the standard way it's much more painful than this: sketchtoy.com/60373153 $\endgroup$ – Shahar Apr 18 '14 at 0:56
  • $\begingroup$ I mean, what you're doing is adding every term in the expansion of the product $(7\cdot 10^2 + 2\cdot 10 + 1)(3 \cdot 10^2 + 4 \cdot 10 + 1)$. The quickness in your usage comes from the fact that you're able to group together the terms of a common power of $10$, and that this happens to work well for you. I'm not sure most would agree that the computation you do is easier than the usual order of adding the terms. $\endgroup$ – Omnomnomnom Apr 18 '14 at 1:01
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Here is a trick to multiply two numbers between 10 and 19 together, say $10 + x$ and $10 + y$: Compute $10 + x + y$, put a zero at the end (multiply by 10), and add $x\cdot y$. Thus $(10+x)(10+y) = 10\cdot(10 + x + y) + xy$.

Easy with algebra. I learned this from my mother who had only an 8th grade education and no algebra.

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