Low Hanging Fruit

I seem to be settling into a practice routine where I am now seeing slow incremental improvements.

I am starting to be able to use the first hundred squares as in calculations.

I am starting know my x^y to the first 5 powers.

I am shifting bases when calculating 2x2 when it makes sense (slowly) and using pretty much all the common shortcuts to calculate generally sub 5 second with about an 80% success rate. I have to double check myself or go slowly for a 98% success rate (more like 10 seconds)

I have been paying extra attention to learning to 20x20 rather than calculating them.

I have been looking at logs, division, roots for amusement but not practicing them in any meaningful way.

I suspect in 6 or 8 weeks I should be using a lot more memory with 2x2 and much less calculation.

On to the question of the evening … Bang for the buck… What will give me the next quickest progress in mental calculation. Am I missing any low cost high return skills I should be looking at?

Thanks as always
Robert :wink:

Concentrate on doing 2x2’s with 100% accuracy.
Find out which ones are the slowest and focus on improving those.

For some this is multiplications like 59 X 89.
Then they find out you can do this quickly by doing 5400 - 150 + 1 (why?)

Use complements to improve subtraction.

The speed you gain with the 2x2’s will also improve your 3x3’s.

Hello Robert,

My good friend Kinma beat me on my main three points but I’ll reiterate them to emphasise their importance (though I’m sure Kinma’s recommendation is more than sufficient for you) -

  1. Concentrate on 2x2 with perfect accuracy before advancing. An important step that many calculators miss, it is much better to take time to master the fundamentals and then build on them rather than having to go back later when you realise your foundation is insufficient. If it helps provide motivation, division is a straightforward process when you’re very comfortable with division e.g. 563/7, you can do this easily since you know 78 is 56 and therefore 780 is 560 then you immediately know that the answer is 80 and 3/7 (I write it this way because that is how you should think of it since decimalising fractions is a far superior approach than actually performing the division each time; not that anyone would do that). I hope the example demonstrates how all this time you dedicate to multiplication will pay dividends with division…

  2. Kinma’s example of 59*89 nicely illustrates the “maths of least resistance” approach every calculator I know of takes. They calculate by whichever way is easiest for the particular numbers involved. You could solve this problem by (50+9)(80+9) = 4000 + 450 + 720 + 81 but that is of course not particularly efficient… (I hope you don’t mind my adding this hint Kinma). Another common way to aid 2-by-2 multiplication, assuming you know your squares or are more comfortable with squares, is via difference of squares. Are you familiar with this?

  3. Complements, complements, complements. I can’t emphasise this enough, I really can’t. It’s beyond an advantage, it’s pretty much essential in the middle of large calculations where you brainpower is a sacred resource. Aside from that, complements make subtractions basically trivial. (EDIT: by which I mean, it makes it lovely ol’ addition)

My interest in mental arithmetic has been on and off and I’m definitely not adept at the craft, but I thought I’d share my opinion.

I have a few books that cover the subject. They are full of tricks and techniques that are supposed to help in becoming proficient at the skill of mental arithmetic. Formulas, tables, shortcuts, and lots and lots of exercises.

I’ve come to think that the last item in that list is the most important. In fact, the rest feels utterly useless for me.

I tried Trachtenberg, Vedic mathematics, memorizing tables etc. (by rote, as I’m not a big fan of Loci). None of that really gave me a sense of progress or enlightenment.

I think the two most important skills pertaining to mental calculation are number sense and short-term memory. I also kind of think that memory is akin to sensing, so I feel tempted to lump those two together and call it “feeling connected with numbers.”

Techniques like the Method of Loci are undoubtedly immensely useful for committing stuff like multiplication tables to long-term memory, but I suspect you don’t need them to perform mental calculation at a level that would blow your own mind. In fact, I’ve found in my own endeavors that often amassing information may be a detriment to deeper internalization and systemic understanding of a subject, i.e. it can detach and prevent you from “feeling” it. This is definitely true for me, YMMV.

Were I to start seriously practicing again, I would invest all effort into developing a closer connection with numbers. It would be the key factor in adopting and utilizing all the tricks and techniques, whereas - in my opinion - the latter is relatively useless in teaching the former. When you know (and feel) single-digit numbers and their relations inside-out and around the block, it’s a firm ground on top of which to build the rest.

If you’re a beginner, tricks (i.e. shortcuts) may do more harm than good, although they do have the benefit of giving experiences of accomplishment and excitement early on to keep you motivated. I just think that most of the tricks are naturally occurring phenomena that reveal themselves to you organically through the progressive deepening of that “direct” connection with numbers. The ones you’d be less likely to discover on your own are much more useful after your mind is ready to fully utilize them. If you just keep crunching numbers you may one day find that you’ve accidentally memorized the multiplication table you never got around to memorizing as if a prerequisite for learning to calculate.

I would even advise against refactoring before the most basic and essential number sense has been developed to a comfortable level. If you feel tempted to simplify the problem, it should be indication that that’s the one thing you should avoid. So if you hate working with digits closer to 10 and carrying a lot, that’s exactly what you need to be doing more, not less.

My advice of course only pertains to learning, not competing. You don’t want to be fast - you want to be thorough, involved and comfortable. Increased processing speed will be a side-product of learning to think fluently. If you only push for speed, you may be neglecting something more essential to actually building up that speed later. Habits may be difficult to break and as far as I understand learning involves both reinforcing and pruning of neural connections, so I would put thought into what kind of practice routine might give my brain the most functional and extensible programming (instead of an island-skill or a gimmick).

If you have trouble visualizing the calculations (as do I), memory techniques will definitely help there, and you don’t even need a massive memory palace (or your preferred mental realm) to retain steps of up to (or beyond) 10x10 digit calculation in memory (assuming you would be using a sensible set of PAO for example).

When I was doing multiplication I tended to work from sides toward the center, i.e. if I were to calculate 59x89, I would proceed like so:

Left: 5x8 = 40
Right: 9x9 = 81

I try to stick these in memory and give them “identity” of residing at the edges. Next I get the number in between:

5x9 + 9x8 = 45 + 72 = 117

Then, in case of three-digit numbers, I would reduce them down to two by moving the hundreds to the tens of the number to the left:

40 becomes 50
117 becomes 17


Then I would visualize the edge pairs 5081 and stack the center pair on top of it:


Then I start reading:

1 0, 1
7 8, 15
5 2 5 1

Of course, this is a terrible way of doing it and you may well be ahead by miles, but it’s something that felt natural to me at the time. It’s a trick for managing the process and a poor one at that. The best I have managed was 36^6 using cubing and squaring in sequence but the speed wasn’t spectacular (I think it took me over 40 minutes!). I kind of lost the motivation after struggling too much with visualization while not having the interest to learn to utilize more robust memory techniques. I also forced myself to memorize squares of numbers up to approx. 60. After all the effort I felt my number sense was as abysmal as ever, and I found myself compensating for it by using tricks. Next time I know better and will concentrate fully on addition and subtraction (complements and carry).

I don’t totally downplay the ingenious tricks taught in some of the literature I’ve read on the subject, but for me they mostly felt too context-specific and gimmicky. I believe that mental arithmetic is more a simple, acquired skill and not an intellectual endeavor. Focusing too much on theory and techniques and too little on repetitive, monotonous drills ended in disappointment for me. Now, I’d leave all those books on the shelf and start grinding column addition instead!

Hi Kannas,

Great to hear how you solve multiplications mentally.
There are indeed a lot of books containing tricks. I think people need to learn a generic method and then a couple of tricks for special cases. With the generic method you can solve anything and with the tricks you can sometimes speed things up.

I was thinking about the way you solve 59X89. Did you ever try to solve it as (60 -1) X (90 -1)?
You can use your own specific method. The difference is less carrying.

If you use your method and work from the side to the center, this is how I think you would do it:
Left: 60X90 = 5400
Right: -1 X -1 = 1

Edge pairs: 5401

Middle: -1X6 + -1X9 = -15

Stack the end pieces onto the middle:


5401 - 150 = 5251

Btw.: congratulation on solving 36^6. This is not an easy thing to do!

I find that learning tricks; using difference of squares, distribution, associativity, factoring, bases, complements, is improving my essential number sense… Albeit slowly. Making real physical changes to the brain takes time as well as correct practice. Brain plasticity is a two edged sword. Rote memorization has its place. I do not imagine 4+5 or 13*12 as calculation. Facts reduce mental load dramatically. Being able to use those facts with fluency is another layer of skill.

There seems to be a range of productive exercises necessary that include memorization of facts, slow thoughtful exercise, fast exercise, integration of methods, mnemonics, pure repetition, practice over time… Trying to pull one method out of the pile as a solution doesn’t seem to help much… Although I have never managed 40 minutes of concentrated effort or calculated anything to the sixth power.

Alternatively starting young when brain plasticity is a given is likely a key contributing factor. On the other hand adult learning is a lot more fun in many ways.

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I think I would normally try to simplify the calculation, but I wanted to demonstrate the raw method. The transformation you propose looks familiar in another context, but I never thought to apply it in cross-multiplication like that. I have also never really internalized techniques like this well, although sometimes I may instinctively use them. The rules usually confuse me more than they help me to process things. That effect is one reason I turned away from trying to learn about methods, and to learning about numbers themselves instead. That said, your insight is valuable to me (as always). I’ll have to try if I can easily augment my routine with this trick.

P.s. Thanks for the pat but honestly the feat was more a display of persistence than anything else. I also didn’t do repeated multiplication but instead first used an algorithm for cubing and then squared the result with the same method as above.


I hope I didn’t come forward as trying to dictate how you should proceed - I only speak from my own experience and framework. Your experience sounds different from mine. For me, the most important aspect is fun. I probably don’t need to tell people how to have more fun.

I am however highly skeptic whether - if time is of essence - anything could be more productive in beginning this particular hobby, than getting comfortable with the simplest things. I started by studying Trachtenberg and Vedic mathematics, and I must say my journey has been highly anti-climactic. Tricks like the ones for doing 9798 (numbers close to 100) or 4545 are fun at first but then you realize how quickly the calculations become more complex if the preconditions are not fulfilled. I felt I was left with a handful of tricks to apply if and only if I got lucky with the operands. I was also doing difficult calculations with very poor tools. Of course I don’t know the specifics on what you have learnt so far (nor what you most value, or aspire for), but judging intuitively I would say that up to 99*99 would be a breeze if one truly knew how to add & subtract.

I think simply doing the thing, the most basic thing; addition and subtraction, dwarfs everything else in importance if you aspire to become a number-juggler. The time one needs to spend on learning, internalizing and applying all the different techniques will be negligible after you can comfortably work with numbers. The most important and valuable shortcuts are not theoretical, but those which your brain constructs by rewiring itself.

Anyone please feel invited to jump in and correct me if you feel that my position is unfounded.

Another point I wanted to make is that many of those tricks are trivial in nature after you’ve actually worked with numbers enough. For example, I learnt from a book on Vedic math that one can do 3535 by prepending 3(3+1) to 25. The formula is then generalized to AB*AC with the precondition B+C=10. Let’s say one is completely oblivious to this cute trick and is using cross-multiplication to methodically grind the numbers the hard way:


AA = 33 = 09
AB + AC = 15+15 = 30
BC = 55 = 25


When I’m doing 2nd powers, I actually do this:

(3 * 5) * 2 = 30 commutativity: ab=ba

When the first numbers (A) are the same (3), what the middle part actually comes down to is:

3 * (5 + 5) distributivity: ab+ac=a(b+c)

Now it is self-evident why if A = A and B + C = 10, and thus A * (B + C) mod 10 = 0, it follows that we can skip doing the cross-multiplication and add A (which is the carry from the middle pair) to the first pair, and thus AA becomes AA+A and the tailing zero means no operation is required on the last pair.

While it may sound overly complicated when spelled out so specifically, it is of course a simple - though curious - pattern and I think many such patterns reach your awareness simply through the act of working with numbers. Depending on your style it may or may not be the 5th, 10th, or even 30th time you crunch those numbers that you figure it out, but once you do it’s much more enlightening and rewarding than just mechanically applying the technique. Also, it’ll be trivial.

I suspect that their are many paths to Rome. I wouldn’t be terribly surprised if time spent practicing basic calculation had a much better correlation with speed and complexity than did selection of training technique ;)…

Between sessions it’s fun to debate the relative merits of the color of the equipment.

Anything that keeps you entertained and motivated enough to practice every day is probably good. If my basic addition and subtraction were better my multiplication and division would both improve…

I have been drilling flash Anzan with both mixed +/- from 1 to 3 digits as well as practicing squares, 2x2, and a little 3x3 the last couple of weeks.

Keeping myself amused/distracted while I become familiar with arithmetic is my biggest challenge. The first 6—12 weeks are always pretty easy but continuing on with daily practice gets dry after a few months if I don’t come up with misguided theories and other random acts of madness.

It would be nice if there were a silver bullet or a tipping point where it all became easy but number of sleeps, number of practice sessions, and slow week to week improvements with the occasional set back seem to be the rule.

Flash Anzan is a simple and effective exercise. I am currently learning the soroban and I use a FA application to practice the mechanics. I do sequences of four three-digit numbers displayed for 3.5 seconds each. I am merely trying to learn to manipulate the soroban unconsciously, and when I’m not physically fiddling with it I try to visualize it and do simple operations.

It is very difficult for me to visualize more than two digits at a time. If the number goes to hundreds, the bead configurations easily escape my attention and fade from memory. I don’t worry too much about the hopelessness of it at this point, and will direct more effort into integrating my other senses into the process (I believe seeing is never just seeing, hearing is not just hearing, and so on…). The tactile dimension is definitely a huge part of using the soroban.

I know what it’s like to get excited about something and then [prematurely] come to a stall and give up on an activity, so what you said about the importance of keeping yourself amused resonates with me. When the progress stagnates I try not to diffuse my focus too much as that pattern trends toward leaving one project in a half-assed state just to move on to the next thing to achieve half-assedness at that. Instead I try to reach out just enough to plug in to those more remote sources of inspiration and insight that the thing I’m trying to get done connects to, while trying to keep at least one foot stationary so that it’s easy to rubberband back and not go wandering.

Switching between systems is a good way to combat the boredom/stagnation. I haven’t tried converting between bases but besides simplifying certain operations it sounds like it could be a good tool for exploration.

It might help (though it definitely isn’t a necessity) if one has at least quasi-pragmatic motives for doing the thing, as practicality implies domains of application and thus convenient ways to test its functionality, track progress and continuously reaffirm oneself to continue doing the thing.

If the thing really is rather arbitrary or one doesn’t know why it should be relevant, it is still worth contemplating why it is. With a stretch of imagination you can turn everything into physics, and relevant.

Skill acquisition proceeds through plateaus. For me, it roughly follows the pattern “struggle, progress, plateau, progress, regress, repeat.” It is that last part, regress, that validates progress to me, as I’ve found that the reconfiguration necessitated by learning new habits virtually wipes the prior rudimentary programming from my brain, i.e. learning is a destructive operation. I now find solace in those moments of sudden-struck ineptness that used to frustrate me before recognizing this pattern, and they only make me anticipate the following quantum leap in skill.

The most difficult phases to deal with are the plateaus, as these are the times when you may feel you’re not making progress, and the plateau phase may last long enough to convince you that you’ve reached your limit, or are approaching the point at which your skill curve starts to flatten toward diminishing returns rendering further investment into honing that particular skill highly uneconomical. At this point one chooses to either stop or continue.

I don’t hold the philosophy that our abilities are limitless, i.e. I think that at the very least the skill curve exists, and therefore it is sometimes the right choice to stop. Of course, we don’t know which plateau is the last, and usually the only way to find out is to suppose the contrary.

what are those important facts that need to be learnt?