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Comment by babuskov | Hacker News Reader

Showing your intelligence at full usually requires that you are not distracted. That is hard to come by these days.

I can calculate 16x180 in my head, but I would still pull out my smartphone to do it if the result is in any way important. Not because I'm lazy, but I know that the device won't make some silly mistake while doing it.

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> Showing your intelligence at full usually requires that you are not distracted. That is hard to come by these days.

The irony that the loss of ability to concentrate is partly a result of smartphones is not lost on me.

it's not "smartphones, it's the lack of adaption as evident from the disparity between generations. The speed of information, the signaltheoretic energy is just steadily increasing and small signals get lost in the noise ...

You accept you might make a silly mistake while doing the calculation in your head. How do you know you won't make a silly mistake when you do the calculation using a machine?

If you do both, or at least estimate the result, you can check the results.

I don't think you'd make a silly mistake when using a calculator, but I think other people might.

I think (although I have nothing to support me) that most people cannot use percentages in any meaningful way. If you say "I have a tv that currently costs $300, and I'm going to give you a 10% discount. What will it cost after the discount?" then they can show you the buttons they'd push to get that result. But if you ask them a slightly different version "I have a washing machine that currently costs $800. I've already given you a 10% discount. How much did it cost before the discount?" then I think you'll find a bunch of people who don't know what buttons to push. (Or worse don't know that they don't know and who'll get a close but incorrect answer.)

So, that's not the kind of mental math trick talked about in the article, but calculators are tools and tools are most useful to people who know how to use them and most calculators and calculator apps don't make it easy for people who don't know math.

What you are describing is high school math. Really simple high school math. Maybe even junior high school math.

I would consider a trick something like, multiply large number by really large number in your head. That is a waste of time.

"Simple high school math" which is sometimes taught in a single hour of a single day and forgotten about by tomorrow because it isn't on the test and we really need to cover the quadratic formula before the mid-term.

Something that I saw on HN not too long ago that really benefited me was something I should have been taught in school. I probably was - but as I mentioned. It was taught in a rush and I had forgotten it, if it was even taught at all.

What is 36% of 25? I'll be honest. I find that's a pretty tricky one to do in my head. I can estimate it to 8 by taking 33% of 24 (close enough, right?). So I'd estimate it to be a bit over 8. Honest guess.

Well the trick I learned on HN is to reverse it. Since % is really multiplication, the commutative property applies. Take 25% of 36 instead and bam! The answer is a flat 9. No estimating needed.

Now the "I'm an idiot" part comes from the fact I always did percentage as multiplication of a decimal. I should have intuitively put 2 and 2 together and figured out the commutative property applies. I hadn't. :)

Given the popularity and support the tip received - I'm going to guess I'm a good example of the average person. The average person was taught percentages in a way that they can convert to decimals but don't think about the math in decimal form. So they miss out on something that is obvious once someone stops and points it out.

So to cut my rambling short: simple math is rarely simple for the average person. The average person, from my experience, is worse at math than they would probably like to admit. (And I'm also an average person.)

Cute trick, but what is 36 * 25? Why is the problem harder when you take out the most trivial part?

It's harder because people fail to make relations or don't think about how they can simplify a problem. Or big numbers are generally harder for mental math. Either or.

Solving 25% of 36 then multiplying by 100 (effectively removing the "%" part of 25%) gets 900. It's also a bit indirect, so many people would overlook it (myself included).

If you were to ask me what 36 * 25 was in most any other context I'd do 40 * 25 = 1,000 - 4*25 = 900.

Although, thanks to your prompt, I'm likely to remember to see if I can cheat with percentages. :) Smaller numbers and fractions are more intuitive for me.

That's a trivial calculation to do, so I think it would be hard to make a mistake. I think you could have provided a better example.

Edit:

To the poster below, you only need the multiplication tables and basic addition. I don't see the big deal.

18 * 10 = 180

10 * 6 = 60

8 * 6 = 48

= 288

I think you overestimate the capabililties of most people.

    16x180
    ...
    = 288

I think he overestimates the capabilities of himself.

He also misidentified the tricky part of the problem as multiplication and addition. Holding intermediate place values is far more likely to cause issues, as we can see from his mistake.

If we allow for simple mistakes, everyone here can do that multiplication in their head without issue. But that was babuskov's exact point. Calculators help with silly mistakes of this sort and they're present nearly everywhere.

I was calculating 18 * 16. For some reason I did not refer back to OP. The approach is similar however.

I agree: holding values is quite difficult in many cases, but the whole discussion is about this example specifically. You only need to remember 3 values in this case, which should be easy for most people.

And no I do not overestimate myself. I usually suffer from imposter syndrome.

12x12 is the max for most memorized multiplication tables (at least in the US). 16x18 is going to require some actual mental work that is prone to errors.

For computer people, 16×18 is easily split into 16×20 - 16×2, which in turn is 320 - 32. If you don't know instinctively that 16×2 is 32 it might be a bit harder though.

Or a better way for us programmers:

16 * 16 = 256

16 * 2 = 32

256+32 = 288

Even better:

16 * 18 = (17-1)*(17+1) = 17^2-1 = 289-1 = 288.

Provided of course you know your squares.

Ah that's a good one. I had powers of 2 in mind.

See my edit. The basic tables are all you need.

  16x18 = 17x17 - 1 = 289 - 1 = 288

Te squares up to 20x20 everybody should know by head.

The only problem is that answer. Seeing 288 (2x12^2) immediately made me spend time double-checking the factors in the multiplicands.

Out of curiosity, in what situations does knowing the squares benefit someone later in mathematics / life? Actually curious.

Short version: it is fun.

If you do mental math, it makes exercises way more fun if you know more interesting numbers. The 16x18 example is an example.

For another example, consider

  42x49

When primed by 16x18, one might consider writing it as

  (45-3)x((45+3)+1) = 45^2-3^2 + 42 = 2025 - 9 + 42 = 2058

(This works for many pairs of numbers that are relatively close together, only requires one 'large' multiplication, and that one is simple, as stated in the article)

But of course, half-way through, one may realize it is

  6x7 x 7x7 = 7^4 - 7^3

Now, 7^3 = 343 (easy to remember in combination with 3^5 = 243), but 7^4 = 2401, for me, doesn't pop out (when you say 2401, I know it's 7^4, but not vice versa)

Meanwhile 49 is close to 50, so we have

  42x50 - 42 = 2100 - 42 = 2058

That is faster, but somewhat dull.

Meanwhile, as far as we know, the savant calculator just takes the dull road, adding 40x40, 40x(2+9), and 2x9, doesn't get distracted, and produces the right answer in a tenth of the time I take. He has less fun, though, because I take detours to visit tourist attractions.

It also can make taxi rides more fun.

Thats not the answer he wrote 16x180=2880

It's funny how he ended up proving that @babuskov was right.

Ah the extra zero makes it so very complicated. I was calculating 18 * 16 and didn't check OP.

That's still a bit slower than it needs to be. The real trick to doing something like is seeing that we have 10(18x16) = 10(17^2 - 1) = 10(289-1)

Having the first 30 squares committed to memory has been pretty useful for me.

Right. And just to be explicit, we're using the fact that (x+1)(x-1) = x² -1. So to add in that step, it would be:

18 x 16 = (17+1)(17-1) = 10(17²-1)

My preferred way of doing this is by subtracting instead of adding:

    16*200=3200
    16*20=320
    16*180=3200-320=2880

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