Help choosing a system for mental math

I apologize for what probably appears as a redundant thread. I actually did read through all the others, but there didn’t seem to be a consensus about which mental math system is best. I’m guessing this is because “best” really depends on what you’re hoping to achieve, so I’ll start there.

I’m looking for a system that allows for rapid multiplication of large numbers (maybe up to 4 x 4) and computation of square roots (and perhaps other fractional powers), and I’d like to do this without looking at written numbers for reference. I’d also (ideally) like the number to be produced from left to right, if possible.

I’m leaning towards Arthur Benjamin’s method (although I’ve never seen him compute square roots), but my concern is that I’ll invest 100 hours or so in one method just to find out that it’s deficient in one particular area, or an entirely different method is superior.

Does anybody know what system Granth Thakkar uses? I saw a video of him (or perhaps it was a similar competitor) doing 20 x 20 multiplication in his head. In the video, however, he looked at the numbers for reference, and he wrote down numbers as they were computed.

My suspicion is that one method may work best with smaller numbers (like 4 x 4) and lend itself more to rapid calculation without looking at written numbers, whereas another system–maybe like the one Granth Thakkar uses–may be better suited for longer calculations but would be inherently cumbersome without looking at numbers for reference.

Sorry for the long post. Any guidance at all would be greatly appreciated.

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IMHO, there just isn’t any system that is best.
Depending on the numbers given there is a generic method and there are shortcuts.

Start with the criss cross method. This is the most generic one.
Then learn the shortcuts. See my old posts for examples.

Arthur Benjamins system is great.
Definitely worth learning an using.

Also learn ‘difference of squares’.

Imo, it is best to start out using all the systems and calculating a multiplication using them all.

So for example:
23 X 37

Arthur Benjamin:
20 X 40 + 3 X 17 = 800 + 51 = 851

Difference of squares:
30^2 - 7^2 = 900 - 49 = 851

Criss cross:
20 X 30 + 20 X 7 +30 X 3 + 3 X 7 = 600 + 140 + 90 + 21 = 851

Also learn to do the 9 proof and 11 proof .
This will tell you wether your answer is correct.

9 proof
23 mod 9 = 5
37 mod 9 = 1
5X1=5

851 mod 9 = 5

11 proof:
23 mod 11 = 1
37 mod 11 = 4
1X4 = 4

851 mod 11 = 4

If both the 9 proof and the 11 proof come out correct you can be certain that your answer is correct.

Also; it builds your mind to be able to juggle large numbers.
Something you need when you get to 20x20.

Let me know if you need more info.
I’ll be happy to explain any of the methods used above.

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Also in my last post in another thread I talk about checking your answer as a way to build proficiency and memory.

A couple of days ago I had to do 359 X 421.
This is an ideal candidate for the difference of squares:
359 X 421 = 390^2 - 31^2
390^ = 152,100
31^2 = 961
152,100 - 961 = 151,139 (first subtract 1,000 from 152,100, then add 39).

Now check the answer using 11 proof:
359 - 330 = 29
29 - 22 = 7

421 - 330 = 91
91 - 88 = 3

Alternatively, if you see 440 quicker than 330 as a factor of 11:
421 - 440 = -19
-19 + 22 = 3

7X3 = 21
21 - 11 = 10

The answer:
151,139 - 110,000 = 41,139
41,139 - 33,000 - 8,139
8,139 - 7,700 = 439
439 - 440 = -1
-1 + 11 = 10

10=10, so the answer is correct.

It is a lot to type out, but in your mind these steps go quickly.
More importantly, it builds the memory muscle you need to do bigger and bigger calculations.

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Thank you so much for the information, Kinma. I appreciate you taking the time to explain things.

This makes sense. I didn’t consider it at first, but in hindsight it makes sense learning multiple methods. One method may lend itself better to a particular calculation, or the problem might just strike you in a way where it happens to fit with one system over another.

I just bought Arthur Benjamin’s book, so I’ll start reading through that. You also mentioned the difference of two squares. I’ve used it a lot in algebra, but I never considered how it lends itself to mental math (cool stuff). It’s definitely a useful tool to have. Would you recommend memorizing squares to increase speed? Right now I just know them up to 15. Also, I’ll read over your old posts for more information.

Oh, and you mentioned the system that Granth Thakkar uses in a PM. Would you mind elaborating on that a bit? Or maybe referring me to an old post? Is it different than what you mentioned here? I’m guessing it’s a combination of multiple methods in addition to shortcuts that only work for certain numbers. I mentioned him because he seems to be one of the most proficient at mental math in the world. Whatever he does, it clearly works very well.

Lastly, do you have any tips for square roots? Maybe this will come up when I search your old posts. I’m going to look right now.

Anyway, thanks again for your time. I really appreciate it.

edit Do you have any comments on the Soroban? Would you recommend practicing it for rapid addition/subtraction (with the goal of ultimately visualizing the Soroban mentally)? I did a little research and the results are very impressive. I also read one of your posts about it.

This is exactly the way I like to work too.

I like to be able to calculate when there is no way of writing things down. Like while driving a car for example.
This is also the most taxing way on the brain and memory.

It becomes more difficult when numbers get larger.
For a 20X20 multiplication it is near impossible. The answer could have 39 digits (!).

Sure, you can use memory systems to ‘store’ the numbers. Still manipulating them only in your head is almost impossible.

However; for 3x3 and 4x4 or maybe even 5x5, this is humanly possible with some practice.

As the numbers get bigger, the chances of shortcuts become smaller. In the end only the criss cross method is able to do the biggest of multiplications.

Try to see the end effect that you know which system is best as a by-product of immensely valuable mental training.

Yes.

Learn to quickly calculate them when not memorized.

As training and in order to memorize them start with 15 squared = 225.
To get to 16 squared, just add 15 and 16:
225 +15 = 240. 240 +16 = 256.

To get to 17 squared, just add 16 and then 17:
256 +16 = 272. 272 +17 = 289.

As always, check your answer. Let’s do the eleven proof:

16^2 = 256.
16 mod 11 = 5 (16 - 11)
5^2 = 25
25 mod 11 = 3 (25 - 22)

256 mod 11 = 3 (256 - 220 = 36. 36 - 33 = 3)

Works backwards too:
20^2 = 400
19^2 = 400 - 20 - 19 = 361
18^2 = 361 - 19 - 18 = 324
17^2 = 324 - 18 - 17 = 289
etc.

You probably know how to square numbers ending in 0 and 5.
Use those numbers as stepping stones or starting points for this iterative process.

It does and af far as I know there is only one system to calculate those large numbers.
I am working on a separate post about this method.

The best method - imho - is this one:

Keep in mind that those people who can do that extremely quickly have worked with the soroban their whole life.

I cannot do that. If I have to add 54 and 33, I immediately see 87 in my mind.

I would say give it a go and see if that works for you. For me, it doesn’t.

Exactly. And as a future engineer, I find it to be a very attractive facility to have. The techniques even lend themselves to integration. After plugging in the limits of integration, mental math could make the evaluation process very quick (I imagine).

That was my suspicion.

You said that you would recommend that I memorize squares. How far up should I memorize them? I read somewhere that 30 is a good point. Would you agree?

That makes sense. Calculate squares using shortcut methods, then step up or down until you reach the target number.

Awesome! I’ll keep an eye out for it.

I’ll probably give it a try. I can already add/subtract quickly, but it’d be really cool to learn to add/subtract 4 or 5 digit numbers in a couple seconds. Hopefully that will be achievable. I’ll probably start with around 20 minutes/day, and then I’ll see where I am in 3 years or so. I guess it could only help.

Thanks again for all your time, Kinma. I really appreciate it. :slight_smile:

Yes.
25 - or a little further - is a good first stop.
The reason is that there are another two shortcuts for calculating squares.

For numbers close to 50 you can use:

25 + x | x^2

Where x is the distance from 50.

56^2 = 25 + 6 | 6^2 =
31 | 36 =
3136

66^2 =
25+16 | 16^2 =
41 | 256 =
4356

43^2 =
25 - 7 | 7^2 =
18 | 49 =
1849

See how easy this is?

For numbers close to 100 you can use:

100 + 2x | x^2
where x is the distance from 100.

97^2 = 100 - 2*3 | 3^2 =

94 | 09 =

9409

88^2 =
100-24 | 12^2 =
76 | 144 =
7744

117 ^ 2 =
100 + 2*17 | 17^2 =
134 | 289 =
13689

Again, see how easy this is?

(the notation ‘|’ is a 100’s separator)

So by memorizing the squares up to 25 and these two shortcuts you can easily calculate all squares up to 125:
0-25: memorized.
25-75: 50’s shortcut
75 - 125: 100’s shortcut.

Apart from the soroban, the biggest way to speed up both addition and subtraction is to work with complements more. This turns an addition into a subtracting and vice versa.

89 + 35 = (89 + 11) + (35 - 11) = 124

Think ‘what do I need to get from 89 to 100’.

53 - 27 = (53 + 3) - (27 + 3) = 26

Another way of doing this, is to think ‘from 27 to 30 is 3, then from 30 to 53 is 23’.

If you work this way, there is no carry and this makes the calculation a lot easier.

With 4 or 5 digit numbers , mentally use the | notation and treat it as 2 2-digit numbers:

4958
2789 + = >

49 | 58
27 | 89 +

First do the hundreds:
(49+1 ) + (27-1) = 76

Then the singles:
(58 -11) + (89 + 11) = 147

Then put them together again:
76 | 147 =
7747

5-digit addition:

12345
67890 +

Mentally split into:

1 | 23 | 45
6 | 78 | 90 +

and it out work from left to right.

On the soroban you basically do the same thing.
Instead of adding 80 you add 100 by moving the third bead up and then subtract 20 by shifting 2 second beads down.
The soroban lets you automate this process. After a while your fingers only move beads without even thinking.

On my phone, I use this one:

In a browser, try this:
http://www.alcula.com/soroban.php

That does make things a lot easier/faster. That’s really cool. I’ve never seen that approach before. And thank you for the tips on calculating squares. Your advice has been really helpful. :slight_smile:

Just thought I’d post with my progress. I know it’s sometimes considered poor etiquette to bump an old thread, but the forum doesn’t seem very active.

Lately I’ve been using the criss cross method for multiplication. I average 5 - 10 seconds for 2x2’s without looking at the numbers, and I average < 5 seconds while looking at the numbers. I’ve also become consistent at 3x3’s while looking at the numbers and without looking at the numbers, although it takes considerably longer without looking at the numbers because the recall process takes time. I’ve also done up to 4x4 squares without looking at the numbers, and up to 4x4 multiplication (two unique numbers) with looking at the numbers.

I recently learned the major system, so I’ve been experimenting with that, too. It’s amazing how much easier it is to recall digits while using it. I can do 4x4’s without the major system, so with it I should (I hope) at least be able to meet my goal of 5x5 multiplication without looking at the numbers for reference.

I also memorized all my squares up to 50 with fast recall, and I’m currently working on 51 - 99. For the squares I don’t know I use the duplex method. And, lastly, I learned the duplex-based Indian root algorithm, although I haven’t practiced it that much. It takes considerably more time than multiplication.

It’s always okay to continue old threads. The forum has new content every day. :slight_smile:

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Hi,

very good to read about progress made. Two points:

When doing the criss-cross method, I more and more tend to calculate the cross first and then attach the remaining figures afterwards. In a 2x2 multiplication, it is working like this, trying to type what is going on in my mind:

23 x 54 =
First 3x5=15 plus 2x4=8, sum is 23

I immediately see that the ones are larger than ten (this is by just looking at the figures, not by calculation)
So 23 becomes 24 without further calculation, then I just attach the 2 (from 3x4=12) to get 242.

Now, only the hundreds are missing, these are 2x5=10,
so 242 gets 1242. Finshed.

You see, I do the mentally most demanding calculation first, and then just add or attach the missing figures from the simple calculations afterwards, as these are the ones we all know by heart (3x4 and 2x5) and thus do not need much mental power. For me, this is even more important when doing a 3 x 3 multiplication. I start with the largest cross and then add the remaining parts at the end and in front.

As for the squares, I recommend to learn all squares up to 99x99.

We all know 1x1 up to 10x10 by heart.
Learning 11x11 up to 39x39 is not that difficult either.

41x41 to 49x49 I do not bother to learn, I can quickly construct them:

Take 15 as base, add the ones digit from the number and attach to that number the square of the 10’s complement of the ones digit (always using two digits here):

42 x 42 = 15 + 2 || (10-2)^2 = 17 || 8^2 = 17 || 64 = 1764

The squares from 51x51 to 59x59 are even easier to construct without memorizing the actual results:

Take 25 as base, add the ones digit from the number and attach to that number the square of the ones digit (always using two digits here):

52 x 52 = 25 +2 || 2^2 = 27 || 04 = 2704

From 61 x 61 onwards I memorize again. Currently I am quite confident up to 70. There are shortcuts for numbers close to 100 also.

What is the benefit apart from using this for the difference of squares technique? When you know the squares up to 100, you nearly automatically know the squares up to 200 as well:

  1. Take the last two digits as base, ignoring the 1 hundred.
  2. Add to this the original number you want to square. This gives you the first 3 digits of the answer.
  3. Square the base number. This is the last 2 digits of your answer (carry, if needed).

Example:

124 x 124:
24 + 124 || 24 x 24 = 148 || 576 = 15376

You see, just a little extra steps, and there they are.

Torsten

Thanks for the post, Torsten. I haven’t seen that method for computing squares before. I like it. It’s extremely fast. I think one day I might memorize all squares up through 999. Until then, though, methods like the one you described are really useful.

And I’ve never tried computing the cross first in a 3x3 (although I have with a 2x2). I’ve always just gone left to right, keeping the partial product in my head. I’ll try it, though. Maybe it’ll improve my time.

Later on this week (sometime when I’m well rested), I’m going to try using the major system to do a 4x4 without looking at the numbers for reference. It’s a lot of numbers to juggle. It’s exactly 8 more numbers to retain as compared to computing the product while looking at the numbers.

I did it! Just now! :smiley: My first attempt at doing a 4x4 without looking at the numbers being multiplied for reference. I multiplied 4326 x 2342 (answer = 10,131,492). It took me a long time (around 6 minutes) because I wanted to make sure that I wouldn’t make a mistake, but I did it. I used the major system to encode the multiplied numbers, breaking each number into a pair of two digits, then visualizing the numbers stacked on top of each other. Then I did criss cross multiplication left to right ending up with 10,131,492, which I verified with my calculator. Originally, I planned to encode the first part of the partial product, but I couldn’t think of anything helpful, so I just focused on the spatial arrangement of the numbers, recalling them that way. Anyway, that’s it. I’m stoked. :smiley:

Next up is a 5 x 5 while looking at the numbers. And honestly, I think this will be easier than the 4x4 without looking at the numbers.

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First of all, excellent progress!

Thorstens options are really fantastic when you are learning the squares 50-99.

I use a similar method for 50-99 if you know your squares up to 50.
The options Thorstens gives are better for their range (because of less carrying).

Mine is wider. Meaning you only need to learn one. And it works basically for numbers 50-150.

Here it is:
x^2 = 100 -2(100-x) || (100-x)^2

An example, let’s use 71^2
100-71 = 29. So 29 is the square we will use.

We only calculate:
100 - 229 || 29^2
Left part: 100 - 2
29 = 42
Right part: 29^2 = 841

Putting it together:
x^2 = 100 -2(100-x) || (100-x)^2 becomes:

42 || 841
now carry the ‘8’:
5041

Of course this is derived from:
(a-b)^2 = a^2-2ab+b^2 = a(a-2b) + b^2
using a=100 gives

100(100-2*b) + b^2 =>

100-2b || b^2

For 101-150, use almost the same formula:
x^2 = 100 + 2(x - 100) || (x - 100)^2

Duplex is great!
I am working on a post for taking square roots using the duplex method and why it is so much faster than other methods.

Thanks :). And Thanks for sharing your method. I’m currently about 70% done learning my squares from 51 - 99. Do you happen to know any tricks for quickly computing squares from 201 to 999? Duplex is fast, but is there anything faster within that range?

The process seems a little awkward at first, but it can definitely get really fast with practice. After the initial setup it’s just duplex, subtraction, then division repeated.

Above 100, you have to be more creative. For example, take 513^2.

Duplex will give you one digit at a time:
D5 | D51 | D513 | D13 | D3
25 | 10 | 31 | 6 | 9

In order to speed things up, you need to do things that result in more digits per step and are still relatively easy to do.

If you know the squares up to 99, then you can split 513 into 510 and 3.
51 | 3 = 51^2 | 2513 | 3^2

Since 51^2 comes from memory and of course 3^2 also, then you only need to work out 2513 and do 2 additions.

Either that or split into:
5 | 13 = 5^2 | 2513 | 13^2.
Now if you do the split like this, the middle part - 2513 - is extremely easy.

More difficult maybe is:
678^2

You could split into 67 | 8 or 6 | 78.
However; alternatively, you could do 7 | -22.
Then the middle part becomes 1422 or 744, which is easy to do.

Thanks for the information, Kinma. I like that method of splitting things up. It makes sense, but I haven’t considered it before.

Apart from the tips above there are the occasional short cuts.
Like 635^2 = 63X64 | 25

Revisiting 678^2.
Another option of course is factorization.

678 is divisible by 2 and 3, so by 6. 678 / 6 = 113.
We can start with squaring 113 and then multiply by 4 and 9.

113^2=
126 | 169=
12,769

Now multiply by 4. Easier than it might look.
Observe that 12,769 = 12,500 + 250 + 19.
Multiplication by 4 gives: 50,000 + 1,000 + 76 =
51,076

Now multiply by 9.
First by 10:
510,760
Then subtract 51,076.

I mentally do 510 - 51 first: 459
Then 760 - 76 = 700 - 16 = 684

Answer: 459,684

In this case 7 | -22 is still easier:
7^2 | -7*44 | 22^2 =
49 | -308 | 484 = (the negative carry can be tricky at first. here are all the steps:)
46 | -8 | 484 =
45 | 92 | 484 =
45 | 96 | 84 =

Answer: 459,684

Another tip.
If a number is close to a round number, let’s say 639, start with the round number - 640 in this case - and make the correction:

So, 639^2.

a=640
b=-1

Start with 640^2.
64^2 = 4096 so just add 2 zeros:

a^2= 640^2=409,600
-2ab = -1,280.
Thus, subtract 1,280 from 409,600 to get a subtotal of 408,320
b^2 = 1, so
a^2-2ab+b^2 = 408,321

And then I hope you see that 638^2 = 408,320 - 1,280 + 4 = 407,044.
Etc., etc.

Just a progress update:

I did my first 6 x 6 today!! I multiplied 265,422 by 326,183 (answer: 86,576,144,226) and I got it right, calculator verified! I did it using criss cross multiplication, going from left to right, chunking the partial product in groups of 3 digits. It was actually a lot harder than I thought it would be. The large span of the crosses results in extensive carrying, so you constantly have to go back and adjust the partial product. Honestly, it was exhausting to do.

So far my PB is 4 x 4 without looking at the numbers and 6 x 6 while looking at the numbers. Also, 3 x 3’s without looking at the numbers are fairly easy now.

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