The Dying Art of Mental Math Tricks | Hacker News Reader

The Dying Art of Mental Math Tricks

blog.tanyakhovanova.com

198 pointsjimsojim131 comments

Is this still a relevant skill-set to be good at these days? I feel like it's a bit like writing cursive or being able to have good handwriting in general. It's just not that useful anymore unless you're in very specialized fields that require you to be good at mental math and even then how many situations are you in where you're put on the spot that you have to mentally solve something? There's probably more useful things to keep in your brain than these types of tricks.

I don't mean to be so negative though. I certainly think it's very fascinating and interesting just from a pure mathematical perspective. However we also have to consider the utilitarian perspective when asking "why" many people aren't good at something like this anymore.

My maths teachers at school, like probably everyone else's, loved to tell us that we wouldn't be able to carry calculators around with us all the time when we're adults.

The joke's on them. I carry a supercomputer with wireless access to the collective knowledge of humanity in my pocket.

More seriously, if we assume we only have limited time and capacity to learn things, these tools surely free us from learning mundane "tricks" and allow us to further explore more interesting subjects. Sure, an understanding of arithmetic is important, so that we can verify our tools are working. But once that's obtained, let's move on.

Yeah, except you look like a prick taking out your smartphone just to do some basic arithmetic.

It's weird to me that a class of people who like to puff about how intelligent they are are so proud of refusing to function at an incredibly basic human skill.

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.

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

> 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.

This is probably where I get bonus nerd points, but even before smartphones, it wasn't a huge hassle to carry a small slide rule. It still isn't.

Mine told me the same thing, and I listened. It turns out you can't think about numbers with a calculator.

Turns out you don't need many numbers to think about math.

Don't you ever take notes? A lot of people still write. I mean pens are still sold at pretty much every convenience store in every country I've lived in.

Learning cursive is so easy it's kind of amazing to me that some people, especially Americans, feel there's no ROI. Legibility on the other hand seems very easy for some and very hard for others.

Which countries have you lived in? I don't think that convenience stores (aka 7-11s, spar, corner-shops, newsagents) in the UK normally have pens.

I take notes in sermons/talks ... on my mobile [cell-phone].

Are you serious? I think almost all small stores in the UK (and most of Western Europe) will sell you pens.

I must be pen-blind ... will need to do research on my way home tonight!

3:3 for having pens - TIL.

Sure they do. They sell lots of crosswords and puzzle books - clearly complementary goods.

Worth remembering that cursive, as taught in America, is different than "joined writing" as taught in the UK. The American version is a bit harder to read and write, particularly the lower case 's' upper-case 'q' and both cases of 'z'.

https://en.wikipedia.org/wiki/Cursive#/media/File:Cursive.sv...

vs.

http://www.cursivewriting.org/joined-up-handwriting.html

In the old days, American cursive was defined by a periodic national competition. So letters would mutate over time as tastes changed. I remember upper-case Q got really weird (like a number 2) for a while. As a kid I just ignored it and did it my own way, which turns out folks could read anyway.

my cursive (in the rare event that i write it these days) is really a hybrid of standard american cursive and print. I avoid the weird capitals in favor of the print versions that I then just join up with the cursive lowercase letters.

Please give details on the first two sentences of your message.

Kate Gladstone DIRECTOR, World Handwriting Contest CEO, Handwriting Repair/Handwriting That Works http://www.HandwritingThatWorks.com

I remember in grade school when we got the latest winning results, we had to write our capital Qs like a large number two. Because it 'looked prettier'. I refused; I closed the circle so it would look like a real letter Q.

Interesting. When I finally went to college after leaving high school, my handwriting started evolving into a joined style. I just figured that I was writing lazily. I suppose it is the path of least resistance as there aren't stops between letters but there isn't anything extra like there is in cursive.

I kind of agree. Most tricks are just a bit of fun. But some of the time when you need to do some mental arithmetic in the real world you're not actually looking for the right answer. It's more about estimating things. No one really needs to be able to multiply a pair of two digit numbers in their head, but if you're at a hardware store and you need to figure out how much paint to buy, being able to estimate the area of a wall quickly having a trick up your sleeve that gets you most of the way is very handy indeed.

Also, a lot of the time these sorts of things aren't going to be used directly, but are more of an indicator of what to type in to a calculator. Knowing how to do something in your head informs how you do it on a calculator even if you don't actually use the trick yourself. A lot of people who are bad at maths don't even know where to start, so having a device to do the actual sums doesn't help them.

I kind of agree with you too. As with most things, it's not black and white. It's a spectrum. And the usefulness has shifted toward the "less useful" portion of the spectrum whereas in the past, mental math was far more useful and the ROI on being good at it was greater than it is in the present.

I'm not suggesting that you be totally braindead and not invest in learning the basics of mental arithmetic. I'm saying that going deeper than that initial basic investment isn't as useful today and doesn't make as much sense as in the past.

For example, I'd consider the 75^2 example in this blogpost very cool, but not worth for me to deliberately remember and so I'm not going to commit it to memory.

Squares are worth knowing because a^2 - b^2 = (a+b) (a-b). This means any two numbers which are close together can be multiplied by finding the square of the midpoint.

For example, 79*71 = 75^2-4^2 = 5625 - 16 = 5609.

If you know your squares up to 100 and your multiplication tables up to 20, you can solve a great deal of two digit multiplication problems without having to reach for a calculator.

The downside to this is if you have an alarm that requires you to solve such a problem, you might solve it while half-asleep and then not wake up.

there ain't no such thing as a free lunch

Although I am not one of "math" people, I think that the pervasive opinion that "what I don't need in everyday situations and work I do not need" is a BIG mistake.

Instead elaborating, I'll just make an analogy: We do not need to run, or to lift weights, nor exercise at all. We have transportation and tools. But, those activities are still highly valuable, even though we do not use them for work nor need them in everyday situations, because our body evolved to need them.

So, math puzzles may not be directly useful, but they prepare the brain for other utilitarian activities. Ok, maybe we do not need puzzles, but math tasks in general is one of the best if not THE best exercise for abstract logical thinking.

Though exercising is not valuable because we need it, it's valuable because it keeps you healthy. I'm not sure how much evidence there is that maths keep you healthy. Sure, if you enjoy it, but if you hate it?

At least exercise keeps you healthy even if you hate it.

It is still good to have basic math numeracy, because swindlers are sadly not a dying breed. It's useful to listen to propaganda and have a feel for whether what the person is saying can even possibly be true.

For example, if somebody says 40 million people in the US are [afflicted with|believe|whatever] something, I automatically think "that's about 1 in ten" and my bullshit filter makes a judgement.

So getting precise numbers is perhaps not so important (maybe it never really was), but being able to estimate quickly it good.

And, FWIW, you'd be surprised how well you can do at the grocery store if you estimate your bill simply by rounding every item to the nearest dollar. Easy to track, comes out remarkably close.

You must not have retired grandparents...

here's an old paper excerpt[1] about the "use-it-or-lose-it" nature of your brain:

> The protective effects of an active cognitive lifestyle arise through multiple biological pathways, new research suggests. For some time researchers have been aware of a link between what we do with our brains and the long term risk for dementia. In general, those who are more mentally active or maintain an active cognitive lifestyle throughout their lives are at lower risk. New research throws some light on what may be happening at the biological level.

[1]: http://www.sciencedaily.com/releases/2012/04/120425094358.ht...

That doesn't necessarily mean mental math is something that will offset that. There isn't much variation in mental math tricks.

> but math tasks in general is one of the best if not THE best exercise for abstract logical thinking.

I think you are conflating 'Math' and 'mental arithmetic'. While the 2nd may help towards the first, it is neither a requirement nor an abstract thinking task on its own.

Maybe not in the described form, but it is still very useful if you can do ad-hoc fast calculations. It helps when talking about budgets and similar - when someone throws some numbers at you it helps immensely if you can check if they are in the correct ballpark.

In the startup world hugely. When you're running a startup your day-to-day life revolves around pricing models, conversion funnels, budgets, commission schemes, etc.

If you're not able to do mental calculations while in the middle of conversations/presentations you'll get left behind by everyone who can.

I can imagine that. Me and my mom are of the calculating kind so when I visit over holidays it's not uncommon we sling a bunch of numbers back and forth over the dinner table to figure something out, and after we're done my dad and sister ask how we arrived at the third number from the first two.

But in more "normal" social contexts, what tends to happen is one of the people involved sit with a digital calculator and input the things other people ask for. It's a bit slower, but the results are just as good.

This falls in the very specialized fields, but traders/brokers certainly need to be good at mental calculations.

At least in electrical engineering, we use mental math every day. More specifically, we use approximate mental math (trying to calculate to within about +/- 25% of the correct value) because it's faster than calculating the exact answer when we need a rough estimate of a power, current, voltage, or impedance number. I imagine that fast, approximate mental math is similarly useful in other engineering disciplines.

Approximate mental math is really good for doing Fermi estimates, too.

One of the most important things I learned from my EE prof was that pi is 3. The square root of 10 is also 3. If you need more exact calculations, you should do it algebraically and then plug in numbers to a calculator, but for quick feasibility/sanity checks, approximate mental math is a very useful tool.

I was taught the importance of this in my first year of Physics undergrad. Fermi approximations (or order of magnitude estimation), from Fermi problems (https://en.wikipedia.org/wiki/Fermi_problem).

It is a highly valuable skill in STEM fields in my opinion. Sometimes knowing a rough number can tell you what to expect, and can immediately indicate whether something is wrong (or could go wrong).

Children in China remember the multiplication table from 1 to 10 at grade 3, as part of the curriculum, as well as additions and subtractions with arbitrary number of digits.

Think of a piece of software where the hot paths get their result directly inlined for perf. Basic arithmetics occupy the "hot path" of your math curriculum until at least high school or beyond. This has a huge influence on reducing the amount of information you need to temporarily hold in your brain at once, leaving room for more important calculations such as exploring the solution space creatively, where one extra piece of information can make or break an insight.

Similarly, you can certainly look up a frequently used word in the dictionary every time if you need to. But you might as well memorize the definition once and for all. Now it's a question of where to draw the line. For creative professions such as math/engineering/art, I'd definitely prefer hard-coding some calculations in my brain for the previously mentioned reason.

> ... leaving room for more important calculations such as exploring the solution space creatively ...

Nicely articulated! And I believe the same applies moving ever higher up. Really, theorems in mathematics could be viewed in this manner -- do the proof once or twice for understanding and hard-code the result for use later in some larger context.

We call it memoization when it's for computers.

People need 0-9 for basic algorithm, 10 is useful for decimals / notation, 11 is easy to learn and 12 is debatable, but potentially handy for base 12 systems which are fairly common. Beyond that we can learn fairly simple algorithms to do this sort of thing.

Then there's the unknown of how it may or may not improve your various brain functions/brain health, like physical exercise is believed to improve your physical health.

Doing things because they might be healthy is a pretty bad reason to do things. See also: alternative medicine.

Being able to quickly sanity check the output of something has saved me from getting fired quite a few times.

I think this is where a lot of people struggle. I can look at some figures and have a rough idea what the result should be, often that is all I need. If I need an exact answer I will work it out properly. Many of the people I work with can't do that - note I'm not in the IT industry any more where I think mental maths was more common.

One example is I regularly get a spreadsheet with product and packaging weighs which we process for levy payments. It could be the weights of the carton, EPS padding, plastic bag sleeves for kettles; all the bits you throw away. My assistant didn't see a problem with a tiny plastic bag weighing 4Kgs, it should have been 4g. That was instantly obvious to me but clearly not to the supplier or my assistant. This is the ability to estimate and get a feel for the figures and although I find it frustrating I accept that many people have a big problem with thinking like that.

I find myself calculating a lot in my head for example when discussing surveys, statistics etc, just to make some quick calc where certain numbers come from etc. Of course you could take out a calculator, but that's cumbersome in the middle of discussions for example.

I dare say that in the various fields within finance, this skill is the true one-upmanship. A good working memory and enough calculatory skills to quickly glance over numbers and models to spot errors and interesting results pretty much defines your worth.

In jest: two types of Excel-users. Those that use the model and those that don't trust the model.

> Is this still a relevant skill-set to be good at these days?

Yes it is! There's even a video

https://www.youtube.com/watch?v=DZCm2QQZVYk&t=35s

"The greatest shortcoming of the human race is our inability to understand the exponential function."

> I feel like it's a bit like writing cursive or being able to have good handwriting in general.

Does writing cursive refer to any writing at all, or do you mean a specific font? Writing in capital letters is sloooooow.

If you cannot write, how would you write on a whiteboard, make mental notes, and so on?

> I feel like it's a bit like writing cursive or being able to have good handwriting in general. It's just not that useful anymore

People teaching hand-writing today concentrate on speed and readability rather than beautiful writing. Some cursive forms are good for speed and readability. Taking notes is still an important skill for students, and computer note taking systems aren't always good enough.

There's not much research, but the little bits that exist suggest that taking notes helps people learn.

Some systems, like Briem, are fast and look neat enough but can end up as hard to read zig-zags. http://briem.net/

This (weirdly expensive book) has details of how writing systems have changed over the years. http://www.amazon.co.uk/Handwriting-Twentieth-Century-Rosema...

I am one of those people good at mental math - enough that our founders turn to me in meetings - and I think it's useful.

Instead of thinking of it as a skill, think of it as an exercise.

There are hundreds if not thousands of things trying to distract, mollify, enrage, and lull us into a "don't think, just act" attitude... having mental fortitude to counteract or ignore those is important.

Mental math (and arithmetic tricks) are pretty important for scoring high on the GMAT (for MBA admissions). No calculator is allowed, and while you can solve by pen and paper, for a very time constrained test, tricks can be the difference between a good and great score.

It's also somewhat valued in the business / consultant crowd for talking through assumptions and making estimates of problems.

when all you have is a hammer, every problem to solve looks like a nail, the more mental tools you have the more likely it is that you'll see patterns or solutions that you wouldn't see otherwise.

The human brain is not a machine that becomes more "optimized" the "less services are running", the more knowledge the better, even in fields that might seem only barely tangential to where your main interests lie

I wonder how much of a kickback Ford and GM would pay for my completely independent iPhone app that gives bad car-buying advice?

If you want to practice skills like these, try:

Street-Fighting Mathematics [1] and The Art of Insight in Science and Engineering by Sanjoy Mahajan (MIT) [2].

Free downloads via: [1] https://mitpress.mit.edu/books/street-fighting-mathematics [2] https://mitpress.mit.edu/books/art-insight-science-and-engin...

Thanks!

This Android alarm clock has dramatically increased my speed and ability to do simple arithmetic: https://play.google.com/store/apps/details?id=com.alarmclock...

It allows the user to set an alarm that can only be turned off by solving math problems. You can imagine how quickly one's brain improves at this task when the reward is a snooze or a removal of a blaring alarm at 5 in the morning.

Has it decreased the median delay between the first alarm and you getting out of bed?

I'd say so. The only way it ever seemed to really work was when I set more than four math problems on the hardest level.

Cool article and I love the writing style with subtle historic comments like the last sentence here:

"""I was good at mental arithmetic and saved myself a lot of money back in the Soviet Union. Every time I shopped I calculated all the charges as I stood at the cash register. I knew exactly how much to pay, which saved me from cheating cashiers. To simplify my practice, the shelves were empty, so I was never buying too many items."""

Reminds me of yet another cool story from Richard Feynman about beating an abacus salesman with mental arithmetic: http://www.ee.ryerson.ca/~elf/abacus/feynman.html

That is a very cool story. Feynman tells through his books a lot of this tricks to do arithmetic.

Mental math is extremely useful when making financial decisions, especially because often you have to think on your feet to decide against an impulse buy and salesmen are actively trying to confuse you. That said schools have never done a good job of teaching finance basics, such as rules of thumb for calculating compound interest, or even what a reasonable interest rate is (hint: inflation is usually 2-3%, while the stock market usually grows at 6-7%).

If you ever need more time "to think and decide" (and feel that you are being rushed) simply ask a question (almost any question will do but one that isn't a simple yes or no is better obviously) or make a statement and use that time to give more thought to what you are trying to make your mind up on.

Stock market annual total return average is 11.41% (1928-2015). Total return means dividends re-invested; not inflation adjusted.

(source : http://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/... )

My favorite trick is one my father taught me after I first learned multiplication in school. The result of any two digit number mutiplied by 11 can be found by taking the sum of the two digits and inserting it between the original digits.

Example:

54 * 11 = 5(5+4)4 = 594

In case the sum is greater than 9, carry the tens digit to the first digit of the result.

Example:

56 * 11 = 5(5+6)6 = (5+1)(1)(6) = 616

I used to challenge my fellow students to see if they could find the result faster using a calculator. They thought I was some genius.

I was wondering why last digit of 5th power is the same as original number and the answer [1] is to look at x^5-x which is x(x²+1)(x+1)(x-1) and prove that it is divisible by 5 and 2 (by going through the possibilities) so it must be divisible by 10. Or [2] use euler theorem. There is a comment that (spoiler) the most general form is "The last digit of, any integer and its nth power, are the same, where n=4k+1."

[1] https://answers.yahoo.com/question/index?qid=20071020225048A...

[2] http://www.johndcook.com/blog/2015/07/04/when-the-last-digit...

I bet this art is not dying, but actually flourishing like never before.

The sheer amount of information in the form of websites, books, videos and so on should be a pointer. I mean just google "mental math tricks"..

It's true that we can offload complicated brain tasks like number crunching to computers, but people do a lot of hard things also because the process gives them satisfaction. They climb mountains and walk thousands of miles not because they want to get somewhere, but because it's exciting and hard.

For this reason, I think people will continue to study and invent new mental math tricks...

More people, more free time, almost infinite info available... No, this art isn't going anywhere ;)

My favorite is still the multiples of 9 by putting down one finger. It amazes me.

Of course it only works with a full set of fingers, but still.

Multiply by x by ten, substract x?

I was wondering why no one was suggesting this, isn't this the most obvious one?

I always found 9 to be easy because whatever you multiply with (2-10) you just had to remove 1 from that number, and maybe memorise the second digit.

e.g.

9*3 = 3-1 = 2 (+ the 7 that I just memorised)

But actually the finger trick is amazing, and I just used it to double check the numbers as it was a loooong time I did it.

The "remove 1" trick is sort of how I learned it, except instead of memorizing "2 goes with 7" I learned that the 2 digits should add back up to 9:

  9*3 = 27 (3-1 = 2,7 = 9-2)
  9*4 = 36 (4-1 = 3,6 = 9-3)
  ...
  9*9 = 81 (9-1 = 8,1 = 9-8)

Fun stuff :)

9s are easy if you know how:

http://www.trachtenbergsystem.org/multiplication-9/

A fun property I learned that helps me remember the 9s: The digits of every multiple of 9, when totaled, eventually add up to nine.

For 9 times x, for 2-10 at least, this can be reverse engineered: The 10s digit of the product will be x-1, and then the 1s digit will be whatever number remains that's needed for the digits to add up to 9.

So 9 times 7: 7-1 = 6, 6+3 = 9, therefore 9*7 = 63.

There may be a similar trick for bigger numbers, but I rarely need to do them quickly so I've never found one.

Two excellent books on the subject are Secrets of Mental Math by Arthur Benjamin (for beginners) and Dead Reckoning: Calculating Without Instruments by Ronald Doerfler (more advanced).

The Mind Your Decisions blog (http://mindyourdecisions.com/blog/tag/mental-math/) has a lot of neat mental math tricks, but they're not really organized into a unified presentation there as in the books above.

I memorized a bunch of these for high school math competitions. You got very familiar with squares up to 100, prime numbers up to 100, converting fractions and repeating decimals. A sample test from back in the day: http://www.texasmath.org/DL/NS/NS9394.pdf Of course, I've forgotten most of the tricks because they were so specific to the test.

> John H. Conway is a master of mental calculations. He even invented an algorithm to calculate the day of the week for any day. He taught it to me, and I too can easily calculate that July 29 of 1926 was Thursday. This is not useful any more. If I google “what day of the week is July 29, 1926,” the first line in big letters says Thursday.

It takes me under 10 seconds to do that one, which is as fast or faster than opening a new window and Googling, especially on mobile, so I think this is still useful.

For the contribution from the year I use my own algorithm that I find faster than Conway's algorithm. Here's mine. In the following, assume a/b means floor(a/b), and odd(k) is true iff k is odd. In a C-like notation, my expression for the year contribution is

   -(y/2 - (odd(y) ? 1 : 0) - (odd(y/2) ? 3 : 0) mod 7

For example, for 26, that gives -(13 - 0 - 3) = 4. The way I would do this mentally is to note that 26 is even so I'm not going to have a subtract 1 step later, divide it by 2 to get 13, note that is odd so subtract 3 giving 10. I then do the negation mod 7 by simply noting how much I have to add to reach a multiple of 7, which in the case is 4 (10 + 4 is a multiple of 7). That gives the final result, 4.

My inner dialog would be "26...13...10...4".

If we were doing year xx27, it would go like this "27...13...10...9...5".

xx28 would go "28...14...0".

xx29 would go "29...14...13...1".

That illustrates all four cases.

I find this simpler than Conway's method, which is (y/12 + y%12 + (y%12)/4) mod 7, although I might find Conway's faster if I would get off my lazy ass and memorize the multiples of 12 up to 100.

I also find it simpler than the odd + 11 method, which is:

  T := y + (odd(y) ? 11 : 0)
  T := T/2
  T := T + (odd(T) ? 11 : 0)
  T := -T mod 7

Odd + 11 has the nice property that you only carry one number of state, whereas mine requires carrying whether the initial year was odd or even. However, it can start out increasing the number you are working with, which slows me down a little with years near the end of a century. Mine always starts out dividing by 2, and then might subtract, so is always going toward lower numbers.

One could remedy this in odd + 11 by changing the first step to the equivalent

  T := y - (odd(y) ? 17 : 0)

when dealing large y values, at the cost of having to do a -17 instead of a +11. (These are equivalent because of the 28 year cycle in the pattern of days of the week within a century. You can start off any of these Doomsday methods by adding 28 to or subtracting 28 from the year. So, if you have an odd year and subtract 28 before starting, and then add 11 under odd + 11, that is the same as subtracting 17 from the original year).

While I'm here, there is one other place that can use improvement. The Wikipedia article on the Doomsday rule gives the rule for calculating the century contribution when using the Julian calendar as:

  6 x (c mod 7) mod 7 + 1

That's fine, but if you just blindly follow it you'll be doing more work than you need to. It can be simplified to this simple expression:

  -c + 1 mod 7

For example, let's do June 15, 1215 (date of the Magna Carta) on the Julian calendar.

Century component: -12 + 1 mod 7. I'd do this by noting that I have to add 2 to 12 to get a multiple of 14 (that's the -12 part), and adding the 1, so I'd mentally just go "12...2...3". The century component is 3.

Year component: "15...7...4...3...4". Year component is 4.

Month component for June is 1, and day is 15 = 1, so we have 1 + 1 + 4 + 3 = 2 = Monday. For the month component, I just memorize it using Conway's suggested mnemonics, which gives 6, but since the month component is subtracted I want the negative of that, and I use the same trick I use everywhere of simply noting what I have to add to reach a multiple of 7. 6 + 1 = 7, so that's where the one comes from.

Trivia: that date is also a Monday on the Gregorian calendar.

A couple other things that might be useful to those wanting to play around with doing calendar calculations in your head.

If you want to go backwards on the year component, and find a year with year contribution M, the first year of the form 4N with year contribution M is (3M % 7)x4.

For example, suppose I want to know a year this century (century factor is 3 for 20xx) where Christmas falls on a Tuesday. The month contribution for December is -12 = -5 = 2. So I want 3 + M + 2 + 25 = 3 mod 7. Thus, I want M = 1. Plugging that into (3M % 7)x4 I get 12, so 2012 has Christmas on Tuesday.

That's already past. I want to know upcoming years with Christmas on Tuesday. We can make use of another pattern to deal with that. The next year after Y within the same century that has the same year contribution is:

   Y + 6  if Y is of the form 4N or 4N + 1
   Y + 11 if Y is of the form 4N + 2
   Y + 5  if Y is of the form 4N + 3

So, starting with a year Y of the form 4N, we have these years all have the same year contribution:

   Y, Y+6, Y+6+11, Y+6+11+6

and then it starts over again at Y+28, which is Y+6+11+6+5.

Using this, and starting from 2012 (a 4N year), we get that Christmas will also be on Tuesday on 2018 (a 4N+2 year = 2012+6), 2029 (a 4N+1 year = 2018+11), and 2035 (a 4N+3 year = 2029+6), and then the 28 year cycle repeats starting at 2040 (a 4N year = 2035 + 5 = 2012 + 28).

An alternative to the (3M%7)x4 approach for going from M to Y is to just memorize this:

  M   First Y for M
  0   0
  1   1
  2   2
  3   3
  4   9
  5   4
  6   5

and use the 28 year cycle to jump up by multiples of 28 if you are interest in a Y for your M that is farther into the century, and use the 6,11,6,5 pattern to move around in shorter ranges.

If you were really using C, that expression could be simplified to something like

    -(y/2 - (y&3) - (y&2)/2) % 7

I also have my own algorithm for computing the day of the week, and I think that you're doing it the hard way. Of course, if it works for you, no problem.

My technique:

  day_of_week = (year_day(Y) + month_day[M] + D - leapyear(Y,M)) mod 7

  year_day(Y) = let Y' = Y-1900 (or Y-1984) in Y' + trunc(Y' / 4)

  month_day[] = { 0, 3, 3, 6, 1, 4, 6, 2, 5, 0, 3, 5 }

  leapyear(Y,M) = 1 if M is Jan or Feb in a leap year, else 0

The year_day() is a constant for any year; in 2016, it's 5. The month_day[] table is easy to memorize in an inverted form that groups all the months that start on the same of the week in non-leap years:

  0: Oct, Jan

  1: May

  2: Aug

  3: Feb, Mar, Nov

  4: Jun

  5: Sep, Dec

  6: Jul, Apr

And that's it. Today (13 Jan 2016) is 5 + 0 + 13 - 1, or 17 which is 3 mod 7, or Wed.

(EDITED to fix line spacing issues.)

For year_day(2016), I get 145 (using the 1900 option). Can you check your formula?

145 = 5 mod 7, and I'll often reduce terms mod 7 when doing the math mentally.

EDIT: I mean of course that 145 is equivalent to 5 (mod 7).

Thanks! I can use this now.

If you use Saturday = 0, like Conway does, instead of Sunday = 0, then your month_day table becomes 1, 4, 4, 0, 2, 5, 0, 3, 6, 1, 4, 6.

Martin Gardner gave a good suggestion for making that easy to memorize. Group it into sets of 3, giving

You can memorize the first three lines as 12^2, 5^2, and 6^2, and then just have to remember that the last line is 2 more than the first.

Here's a cool site for practicing mental math (only 4 operations involving numbers up to 12 though): http://www.speedsums.com

Naturally there is a Dover book that covers this topic too:

http://store.doverpublications.com/048620295x.html

"I was good at mental arithmetic and saved myself a lot of money back in the Soviet Union."

This reveals a weakness in western education. People in the former Soviet Union learned a lot of mathematics in the traditional way (a mixture of clever mental calculating skills and a good grounding in theory) simply because they had no alternative and there were many problems to be solved requiring higher math skills.

Americans, on the other hand, (as well as being famously lazy about math) now rely almost completely on calculators and computers and are gradually abandoning both mental calculation and deep theory in mathematics. An EMP will someday zap all our electronics and we'll get our comeuppance, and we'll have to try to remember exactly why the integral of x^2 is x^3/3 + constant.

As I sailed solo around the world 25 years ago I spent a lot of my free time practicing mental math. One day in Israel I read my receipt at a restaurant and realized the waitress had inflated the cost by performing a creative kind of addition. I corrected her figures, lectured her, and paid the difference as a tip. It occurs to me that she could get away with creative addition with 99% of people, even well-educated people, because nearly no one checks the addition on a receipt.

Hans Bethe and Richard P. Feynman were both formidable mental calculators, at a time when that skill was valuable. There are stories about how (when they were both at Los Alamos) they would have contests to see which could produce the quickest result using somewhat different methods. This was at a time when a matrix calculation required a roomful of "calculators" sitting at mechanical adding machines all day long, sometimes for weeks, estimating the yield of a nuclear explosion.

>and we'll have to try to remember exactly why the integral of x^2 is x^3/3 + constant. //

I could never remember such things when I was younger (it's worse now), which was why I loved maths so much because when I couldn't remember - provided I could remember basic principles - I could work it out.

So as long as we know the basic principles of integration we won't need to remember that particular sum.

I am terrible with remembering a bunch of facts, but I'm pretty good at working things out from basic principles. It's funny because many of my friends look at me figuring something out from basic principles and they think it looks like so much work – then they go back to taking a few hours to memorise things they could just as well work out from basic principles! Who's doing the work now?

I guess we both are, only I postpone it to when I really need it.

Let's play Numberwang! http://youtu.be/qjOZtWZ56lc

If you find Speed Maths interesting, you'll probably want to check out Trachtenberg's "Speed System of Basic Mathematics" book:

http://www.amazon.com/Trachtenberg-Speed-System-Basic-Mathem...

I didn't read all the comments, so this might have been said already. But I think there are some professions where quick mental arithmetic provides an edge. At one point I was learning how to play professional poker and while I had the common situations memorized, I'd calculate the less common situations. There are more use cases than just poker, chance estimation in general, or in a casual discussion where the other might not have a lot of patience.

Compared to the old days however it has been in decline, but just like writing -- even cursive writing -- it still has its uses (cursive writing: exposing yourself to multiple ways of writing allows for a more fluent writing style in my experience).

The First Sunday Doomsday Algorithm is the simplest way to calculate the day of the week for any date in your head.

http://firstsundaydoomsday.blogspot.com

It brings together a bunch of great mnemonics, like the new "odd+11" rule for calculating the 2-digit year code, as well as Conway's classic "I work 9-5 at the 7-11" for the month code.

like any science, math is more about problem solving skills versus the rote action, and mental math nowadays is nothing more than a 'fun' game. However some insight can be gained from mental math tricks that help solve deeper problems (log addition versus product computation for bayesian nets to avoid overflow), but the cost of teaching these tricks conveys to students that the trick is the goal, missing the point of mathematics as a science. At worse teaching such tricks results in us losing otherwise amazingly talented future mathematicians because they weren't particularly good at the 'tricks' part.

Anyone who has undergone a gauntlet of "cargo cult" style technical interviews knows full well that these "tricks" arent a dying art - they sometimes determine eligibility for a job.

Looking at this thread there is clearly a community (overlapping significant portions of the HN community) where this is not a dying art.

Are there any known resources for learning math tricks like the ones mentioned in the blog post? Perhaps a book...

Secrets of mental maths is one book that I heard of a long time ago.

Note: didn't read it myself.

Indeed. In my UK primary school in the 1960s, 'Mental Arithmetic' was a distinct curriculum subject.

It took me a couple seconds, but I did this mentally:

75^2 = (70 + 5)^2 = 4900 + 2 * 70 * 5 + 25 = 5600 + 25 = 5625

You can also do this by (ab)using the "square of a number close to 50" trick. Take 57^2 for example: this is the square of 50 plus 100 times the difference of the number from 50 plus the square of the difference of the number from 50, which is 2500 + 100x7 + 7^2, or 3249. For 43^2 it would be 2500 + 100x-7 + (-7)^2, which is 2500 - 100x7 + 49, or 1849.

With 75, that's 2500 + 100x25 + 25^2. Hopefully you've memorized that 25^2 is 625, which makes this an easy 2500 + 2500 + 625, or 5625.

You're saying that

  x^2 = (50 + (x-50))^2
      = 2500 + 2 * 50 * (x-50) + (x-50)^2
      = 2500 + 100 * (x-50) + (x-50)^2.

I'd be a little worried about the number of double-digit squares in this method. I can't do them as easily as single-digit squares, which using the multiple of 10 closest to the number ensures.

For example, I'd have to use another trick to calculate

  13^2 = 100 + 2 * 10 * 3 + 9 = 169

in the middle of the calculation for

  63^2 = 2500 + 100*13 + 13^2.

But

  63^2 = 3600 + 2 * 60 * 3 + 9 = 3969

is almost immediate.

That's why it's an abuse of the "number close to 50" trick. It really is meant for single digit differences from 50. The only reason it works for 75 is because hopefully 25^2 comes up often enough that people have it memorized. I know I memorized it long ago after seeing it a number of times: "twenty-five times twenty-five? six twenty-five" is easy for me to remember. If people don't have 25^2 memorized, the trick kind of falls down.

Otherwise, you're right. You can break down the square of any number into (x+y)(x+y) = x^2 + 2xy + y^2. For 50, the "2x" term equals 100, and multiplying by 100 and adding to 2500 is dirt simple for anybody to do.

96 81 676761

More generally,

    (n x 10 + 5)^2 = 100n^2 + 2 x 5 x 10n + 25 = 100n^2 + 100n + 25 = n(n + 1) x 100 + 25

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