# How can I start mental calculating

#22

I haven’t read the other responses but the Trachtenberg method of speed mathematics is excellent. I bought a used copy but I believe you can access the PDF version free online.

(Nicholas Mihaila) #23

I remember having this same thought before, how an 8x8 could be reduced to a 4x4. The prospect of memorizing all 2-digit multiplication tables is daunting though to say the least. Do you know by chance if Willem memorized the tables using mnemonics? Or was it just normal spaced repetition? I guess if you memorized 1,000 digits/day you could learn the tables in about a month. I think it’s the retrieval process that would be tricky.

I just wanted to quickly interject that I think the duplex method of roots provides a better algorithm. I read through the Trachtenberg book a while ago.

#24

Keep in mind that although you can change a 8X8 into a 4X4 by using 2 digit numbers, the additions also become much longer.
So it is not certain that you gain time that way.

I know Willem personally, have been to his house many times.
He explains that he remembered the 2x2’s just by doing them (as a kid).
Now he just knows them by heart.
I know; I envy that too.

(Nicholas Mihaila) #25

Absolutely. I’d be interested to know what the time benefit is. I’m sure there’s a way to get a good approximation. You would just have to add the times from each step.

That’s really cool. I wish I was more productive as a kid. I could have been learning my 2x2’s instead of playing all those video games.

#26

What are you now focusing on?
Multiplication? 2x2? 3x3?

Actually now I’m focusing in multiplication in general and particualry I went a little deeper with squaring problems, its way more interesting knowing that arthur benjamin is actually using the “major system” in his calculations processes! And he also know all the squares of all the numbers between 11-99 by heart! Making squaring a 4 digit number very easier adding to that if he is using the major system as a memory devise the helps him holding all these numbers in his mind vivid while multipling/summing them a very interesting way to do it actually I knew it , the key for mental math is how to make these numbers vivid for you while calcualting them.

#28

This is from How to be Clever:

Take the multiplication one digit at a time, starting from the end. For the sum 396 x 184, we know the final digit of the answer will be the final digit of 6 x 4 – which, if you know your times tables you’ll know to be 24. So our last digit is 4 – write it down or remember it, depending on how hard you want to make things for yourself, and carry the 2 into the next calculation.* Now for the second-last digit. It’s less obvious, but this one is made up of the tens from the previous calculation (that 2 we just carried), plus 6 x 8, plus 9 x 4. That is, we’re multiplying the last digit of each number with the second-to-last digit of the other number. 2 + 48 + 36 = 86.

Once again, we take the last digit of the answer, write it down in front of the 4 we already got, and carry the 8. Or, to look at it another way, we’re taking the first number (396) and gradually multiplying each digit by each digit of the second number (184), working backwards. Carrying on in the same way, our third-fromlast digit will be 8 + (6 x 1) + (9 x 8) + (3 x 4), which all adds up to 98.
(…)
Now, we’ve already multiplied the 6 in 396 by each digit of the 184, so we can forget about it now. We’ve carried the 9 from the third calculation, and we add to it 9 x 1 and 3 x 8, giving us 42. We should now have an answer that ends with 2864, just waiting for one last calculation. Carry the 4, and add 3 x 1, and we’re done. 396 times 184 is 72,864.

#29

Funny thing.

The most difficult seeming calculations usually have an easy way of getting to the answer if you think about it longer.
Because I would do the previous calculation as follows:

396
184
_____ X

In my mind I see:

400 - 4
180 + 4
_____ X

Now do criss cross multiplication. From left to right.

400 X 180 = 72,000 (in my mind I do 4x18, then double the first and halve the second number = 8x9 = 72. Add zeros.)

4 X 400 = 1,600
-4 X 180 = -720 (same calculation as the first step}
-4 X 4 = -16

Putting it all together.
1: 72,000
2: 1,600 - 720 = 1,600 - 800 + 80 = 880.
3: 880 -16 = 864. (880 -20 +4)

72,864.

#30

Also observe that the calculation is done without carrying.

#31

Even if you don’t know these, you can easily move from one (known) square to an unknown one.

An example.
Let’s say you need to know 24^2 and you know 25^2 = 625.
We use: (x-1)^2 = x^2 -2x +1
in this case (25-1)^2 = 25^2 -2*25 +1 = 625 -50 +1 = 576.

Going upwards, use (x+1)^2 = x^2 +2x +1

Starting with 20^2=400:
21^2 = 20^2 + 40 +1 =441
22^2 = 441 + 42 + 1 = 484
23^2 = 484 + 44 + 1 = 529
etc.

See how quick you can do this?
Use this as an exercise while waiting for a bus or so.

#32

Squares of two digit numbers ending in 5 can be easily calculated in this fashion:

1) they all end in 5 x 5 = 25 (right hand side)
2) multiply the tenth digits by itself-plus-one (left hand side)

e.g.,

15 → 1 x 2 = 2 … 225
25 → 2 x 3 = 6 … 625
35 → 3 x 4 = 12 … 1225

in fact in general, you can use above method if the tenth digits are the same and the unit digits add to 10…

28 x 22
left hand side: 2 x 3 = 6
right hand side: 8 x 2 = 16
result: 616

73 x 77
lhs: 7 x 8 = 56
rhs: 3 x 7 = 21
result: 5621

The most essential Memory techniques for Math Students - Summary Thread
#33

I’d argue that most people use some kind of number-to-image system when doing mental math with a few more significant digits that require keeping track of. That could be the major system, number shapes, or what have you… most people doing blindfolded Rubik’s cube use a double letter system with speffz. So whilst interesting maybe, that he picked the major system, it’s really not that surprising.

Okay, so there’s 89 squares that he knows (11-99) but did you ever try to structure them in any kind of fashion to see any patterns?

• 20² = 40
• 30² = 90
• etc.

• 15² = 225
• 25² = 625
• etc.

I explained squares ending in five in my previous post. So you already know 17 out of 89 squares, which is almost 20%

• 99² = 9801
• 98² = 9604
• etc.

Here you’re really close to the base 100, so you just subtract the distance from 100 for the left hand side and then multiply the difference for the right hand side. Here with 96 x 96 as an example:

96 - 4
96 - 4

left hand side is 96 - 4 or you could do 100 - 4 - 4 to get to 92. The right hand side is 4 x 4 = 16 and put together for the answer you get 9216. Might be easier to look at 8 x 7 to make things more clear:

8 - 2
7 - 3

Using 10 as a base (instead of 100) you get either 8 - 3 across or 7 - 2 across or 10 - 2 - 3 for a left hand side of 5 and 2 x 3 = 6 for the right hand side… answer 56.

• 11² = 121
• 12² = 144
• etc.

Similar to what we just did, but now adding instead of subtracting. Here with 13 x 13 as an example:

13 + 3
13 + 3

For the left hand side it’s 13 + 3 or 10 + 3 + 3 for a total of 16 and for the right hand side it’s 3 x 3 = 9 and put together 169. You now know 33 of 89 squares or close to 40%

Case 4

• 21² = 441
• 22² = 484
• etc.

Same example that @Kinma gave above but here without recursion… let’s do 23 x 23:

23 + 3
23 + 3

The left hand side is again 23 + 3 or 20 + 3 + 3 for a subtotal of 26. “Subtotal” because our base 20 = 2 x 10, so we need to double 26 now for a total of 52. The right hand side is simple 3 x 3 = 9 (don’t double this). Put together you get 529. One more with 28 x 28:

28 - 2
28 - 2

left hand side: 28 - 2 = 26 subtotal. Using 30 instead of 10, so triple the subtotal 26 x 3 = 78 and the right and side is simply 2 x 2 = 4. Putting them together 784.

I’ll leave case 5 for you as an exercise, but you can image that instead of five times 10 for squares close to 50 you could also use half of 100 instead. Generally, you’d know say 30 from 3 x 3 and 40 from 4 x 4 as well as 35 as 3 x (3 + 1) & 5 x 5, so use base 20 for 21, 22, 23, 24 and base 30 for 29, 28, 27, 26.

You can mix and match of course and for 71 x 71 the method that @Kinma suggested will probably be faster. There’s a few more shortcuts but this will have to do for now… hope you can see that “knowing” all the squares from 1 to 100 is not as big a deal as it would appear at first. Of course if you want, you can always memorize a handful that take you too long to “calculate” this way.

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#34

I mentioned a few more shortcuts in my last reply, so here’s one. Let’s say you’ve made it up to 25² using the methods I’ve mentioned previously. There is a nice symmetry at 25 once you get there:

Use 25 as your base… Δ is the difference to 25. Say you want to square 28 in which case Δ = (28 - 25) = 3; so just take the square of (25 - Δ) = 22² = 484 and add to it (Δ x 100) = 300. The result is 484 + 300 = 784

## Example: 32²

Δ = 7
→ (25 - Δ)² = 18² = 324
→ add Δ x 100 = 700 to 324
and you get 1024

So once you know the squares up to 25, you automatically know the squares up to 50 by reusing the squares from 1 to 25 and performing basic addition. Of course, still use case 0 and case 1 for 30, 35, 40, 45, and 50.

New Member: Andrew, US

Okay, that’s is a great method for calculating the squares of 36 numbers in total between 11-99…!

I use the same method mentioned here …

Very specific I guess!

This is a very nice post! @bjoern.gumboldt , Actually I’m using the same techniques you are describing now, it’s like a summary of all what I got, and no I’m not thinking that knowing all the the squares from 11-99 is kind of a big deal, but what I mean by that is some ppl like Arthur and many others doing mental calcualtion for so long, and with time they just know all the squares without actually calculating them with any tricks/tips because they are familiar with the answer already, if you understand what I mean… it’s just a matter of time… and I guess that I’m heading the right direction with the right tools to make it!

#38

You can continue this idea of symmetry for 50+ now that you got a way to get the first 50 squares. Simply subtract 50 from your given number and add the result to 25 for the left hand side and square it for the right hand side:

## Example: 57²

Δ = 7
→ lhs: 25 + 7 = 32
→ rhs: 7² = 49
and you get 3249