The best Methods Mental Math

Mental Calculation, Mathematics, and Puzzles
1

Hello all…
My methods for Addition , multiplication explained here:

the addition is basis and grouping numbers works well for me
for example:
1 5 4 8 5 9 3 9 5 4 4 5 4
i try to group numbers , groups of values 5 10 15 20 …
first 3 is 10
next 4 is 25 so we have 25+10
i see 4 times 3 so 35+34=47
and 25+9 we have 47+19=66

there are many videos from Youtube and others for multiplying 2 digits numbers
2458 = (25)(45+28)(84)=1392
but it’s hard for 3 digits numbers(or say better it takes time) i am sure even Arthur T. Benjamin never use such methods while those methods rarely used.
when some one asks you to answer 5479 the best time for you is 5 seconds and the maximum time is 10 seconds.
by memorizing different methods of multiplying for different numbers and choosing the best one when some one asks you a question takes more than 10 seconds.
so prodigies in mental math never use these methods like:

8998=(98-11)(112)=8722
11 2
the brain is the same!
good methods and hard training makes them a good calculator.

one of the questions in World mental calculation 2010:
56044527
·10846910

607909940361570

i have done this in 20 minutes!(with the best conditions and full concentration!)
it was interesting for me that my answer was correct! by calculating and memorizing numbers from right to left but the time kills the questioner as well.
it’s like this:
your good friend asks you to calculate the 1548*6231 and you tell him: hey! please wait
some moments(each moment = 2 minutes) and drink a cup of coffee and watch a football game on tv then comeback to me happily and i will give you the answer :slight_smile:
even it’s more disappointing to tell him not exactly but maybe the answer is 9645588

any one with a good method of multiplying please share it!

2

In both his book and video Arthur Benjamin discourages cross multiplication for mental math and leans toward simplifying problems into addition and subtraction when possible.

In the examples you gave …

5479 could be 548 add zero and subtract 54.
54*8 —> 432 add 0 —> 4320 - 54 = 4,266

8998 could be 89100 and subtract double 89.
89*100 —> 8900 —> 8900 - 178 = 8,722

For 2458 you could half one and double the other until you get a single digit multiplier.
2458 = 12116 = 6232 —> (623), (62) —> 138, 12 —> 1,392

If anyone would like a new factory sealed copy of Arthur Benjamin’s Secrets of Mental Math I have a couple for sale.

3

well… i have a gift for speed in mental math. I can multiply two 8 digit numbers in as little as 40-50 seconds, but usually takes me around 70 seconds but the problem is that i make mistakes around 50% of the time. If you don’t believe I can do this, let me know and I’ll make a video to prove it.

4

Hi Martin.
Since you mentioned World mental calculation 2010, I participated in that great competition in Madgeburg. 8x8 digits is the standard format for multiplication.
I solved 4/10 8x8 multi correct in 15’. But since then I impoved. (e.g. 7 correct in Memoriad’12). But the thing is, if you lose 1 digit out of the answer (15 or 16 digits in total) the whole task counts as wrong (0 points). So accuracy is more important than speed.

@ericrulz What’s your record exactly in seconds?
Talking of speed, Mine is 30" for 8x8. check my screen capture: http://i.imgur.com/C491d15.jpg
(of course that was an easy task, that’s why I made a personal record, sometimes it’s a matter of luck to have a few zeroes in the end)

The fastest I know is my friend Freddy from Cuba, he has done one such task in just 13". I think he is faster than what Sakuntala Devi was. But now she’s dead so there is not way of comparing them. But Freddy is the faster in the world in multi, even faster than the Indians.

Nodas

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5

hey nodas … what method do you use?

6

Hi Sharad and greetings to Nepal

I use the criss-cross algorithm. You can find more information here or even more thoroughly at Robert Fountain’s and Jan Van Koningsveld’s book: The Mental Calculator’s Handbook

Nodas

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How can I start mental calculating
7

thank you nodas… i use criss cross too .i can do three digits by three digits but its really hard for me to constantly visualise the numbers im multiplying… i am hopeless about me ever doing four digits by four digits… can you give some ideas about how to develop some more visualising in this aspects thank you again

8

Sharad, my ideas are:

  1. Never say ‘You’re hopeless’. The brain can constantly improve if you have a positive mood, a motivation to get a “growth mindset” and a willingness to break your own limits. In Dresden, I’ve witnessed an Indian kid who was doing a mental multiplication of 20 digit by 20 digit correctly, in less than 3 minutes. But again, that kid was Granth Rakesh Thakkar and he was the Mental Calculation World Cup 2014 overall winner.

  2. After deciding that you want to improve, then the next and most important thing is to start training and practicing. Visualization comes naturally after practice.

  3. Compare just to yourself and your own records. If you compare to others, (and unless you hold the global WR), then there will always exist someone with better records than yours, which might let you feel down initially. But if you compare to yourself only, then the only thing required is to break your own records. That’s a motivation to keep going.

P.S For 3x3 multiplication, in example, the unofficial world record would probably be around 2 seconds on average. That record could be achieved by the best mental calculator in the world : Mrs. Lee from South Korea, whom I’ve witnessed to do so (and also a couple of Japanese guys can also go as fast as her)

Thanks for reading,
Nodas

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9

How do you mentally keep track of the numbers while you’re working?

Example:

\begin{array}{r} &798 \\ \times\!\!\!\!\!\!&923 \\ \hline \end{array}

3 * 8 = 24

Place the 4 and hold the 2 in memory:

\begin{array}{r} &798 \\ \times\!\!\!\!\!\!&923 \\ \hline &4 \end{array}

2 + (2 * 8) + (9 * 3) = 45

Place the 5 and hold the 4 in memory:

\begin{array}{r} &798 \\ \times\!\!\!\!\!\!&923 \\ \hline &54 \end{array}

4 + (9 * 8) + (7 * 3) + (9 * 2) = 115

Place the 5, carry the 11:

\begin{array}{r} &798 \\ \times\!\!\!\!\!\!&923 \\ \hline &554 \end{array}

11 + (9 * 9) + (2 * 7) = 106

Place the 6 and carry the 10:

\begin{array}{r} &798 \\ \times\!\!\!\!\!\!&923 \\ \hline &6554 \end{array}

10 + (7 * 9) = 73
Place the 73:

\begin{array}{r} &798 \\ \times\!\!\!\!\!\!&923 \\ \hline &736554 \end{array}

It seems like it would be more difficult with 8+ digits. Are you using locations or images to hold those in memory, or is practice enough to juggle that many numbers at once with a running total?

Also, is my example the same way that you do it?

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10

thank you alot nodas… im all pumped up now all thanks to your very motivational writing… wish you very best for coming competitions. gotta start practicing and hey greetings to Greece too.

11

Hi Sharad,
The important thing to keep from this thread is to benchmark and measure yourself through timing. This is the M (=measurable) part of the S.M.A.R.T. “goal system” and probably the most important. So, if you can now do a 3x3 mentally in 5 minutes, then next week try to do it in 4 minutes. Next month, in less than 3 minutes and so on. The actual record numbers do not matter as much as the ability to achieve a constant progress. Eventually you can do it less than 1 minute and even less. You can use a timer and stopwatch like those rubik cubers use. Then keep measuring your progress, through screenshots or data sheets. Everyone can break their own records. After reaching some plateau, then progress will stop. But most people have not even reached their own limits.

Hi Josh,
I only use images in recitals when I am not allowed to write or type anything. But in normal mental calculation when writing the direct result ( in MC World Cup) or when typing the direct result in the program (i.e Memoriad software), there I do not use any images, because all the carry numbers are rather small and always either 1 or 2 digits. The first operation is always to get rid of the carry by adding it to the first product of the next product.

I.e. for 99,999,999 x 99,999,999 I don’t even have to remember any particular part of the result 9,999,999,800,000,001, at any point nor any number I wrote or typed.

The the only numbers I use for the above operation are:
81 carry 8; 162 carry 17; 243 carry 26; 324 carry 35; 405 carry 44; 486 carry 53; 567 carry 62; 648 carry 71; 567 carry 63; 486 carry 54; 405 carry 45; 324 carry 36; 243 carry 27; 162 carry 18; 81 carry 9.

Like it happens in the L1/L2 cache in CPUs, one human can also only process very few number blocks at a given time. (in Flash Anzan, around 5 or 6 digits maximum). This digit span is not even a normal short-term memory but something like “instantaneous cache buffer” (or “millisecond short-term memory”)

Anyway, in a random 8x8 multiplication the maximum carry is 71 (i.e. the middle cross-product of 99,999,999 x 99,999,999 is 8x81=648, plus 62 carry from the previous operation, therefore 710). Then you immediately add 71 to the next cross product, therefore 71+7x81 = 638, (type 8, carry 63 and so on).

My preferred format is from “right to left” with a vertical arrangement, as most mental calculators prefer. Mrs Lee, (current multiplication WR ) however, uses left to right (vertical). I’ve seen some people like Hua (Canada/Malaysia) who uses the
horizontal arrangement: 99,999,999 x 99,999,999
instead our normal arrangement of
99,999,999
x 99,999,999.
The former resembles an abacus/Soroban with 16+ positions so I understand why they prefer that.

Bottom line, in recitals the multiplication method is totally different and that’s why you have probably seen Arthur Benjamin to invoke some images when he does 5x5 multiplications in presentations. I also have a small image system (hybrid-Major based, if needed ) but it only works with a few digits and it’s nowhere near as vast as of that of the top memorizers in this site like Alex, Lance or Simon or any of the XMT top competitors.

Nodas

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12

Thanks for the tips. I’m most interested in doing it in my head and reciting it out loud, since I won’t be competing.

I’m wondering where in their minds people place or hold their images. Is there any kind of journey? If so, are they placed in a journey that moves from right-to-left or left-to-right? Or are they visualized in some kind of abstract space?

If one had to recite (not type) 8943 * 9397, the result is eight digits (84037371) – probably longer than most people could hold in mind while doing the calculations. And if the audience speaks a problem instead of writing it (“8943 * 9397”) those numbers have to be stored somewhere too. I curious about those fine details. :slight_smile:

13

Hi Josh,
For mentally doing 8943 * 9397. My steps:

A. I do these two easy 2x2 in my head
89x97 = 8900 - 3x89 = 8633 (holding this on memory) while simultaneously doing 43x93 = 4300 - 7x43 = 3999
( note that 3x89=267 and 7x43=301 are pure memory recall for me)
I’m mentally adding 8633+3999, to immediately get 12632,
which is my carry, which I split into 126 32.
Time needed : about 12" sec. total : 4" sec. each multi, plus 2" for the addition, 2" for carry input.

B. 89x93 = 91 squared - 2 squared = 8277, again that’s pure memory recall, since I know all 2 digit squares by heart. Only a small subtraction was required. Time needed about 5" sec.
Then 8277+126 (my first carry) = 8403, so I gotta remember 8403 32. Again, about 5" sec to register the carry or associate it with some image. IMPORTANT: Forget the partial multiplications, after you do them. Only remember the carry. Psychologists say can we remember about 7 items, so that 6-digit “carry” is still within the theoretical human limit, even without using any image. But if this sounds hard, then 6-digits is just 2 associated items in your 1K P.A.O. system.

C. Last step 43x97=4300 - 3x43 = 4171. Total time: 3" sec since again 3x43=129 is memory recall.
Total time around 25", and finally recite the correct result by combining 8403 32 plus 41 71, so as to correctly utter 8403 73 71. In competitions, I’ve met some Top mental calculators who can do 4x4 recitals in 10 seconds instead of my 25", with no need to write any intermediate results.
The carry here was just a small string of digits. So in my case, a memory palace or journey was not needed. Only just a few algorithms. In paper or in a software like Memoriad, this 4x4 due to the criss-cross method can be solved faster. There are some people who can mentally calculate and simultaneously write(or type) the correct result for the above 4x4 operation, in just 5 seconds. But I find the recital (without writing anything), to be like a more pure ‘mental calculation’ and more fun because it’s more creative than the mechanical criss-cross method. Both categories have their tricks and nuances though. It’s a matter of practice and NOT magic (or “mathemagic”).

Nodas

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14

Thanks for the details. Did you intentionally memorize those, or did they get memorized simply though repetition?

15

Hi Josh,

You nailed it in both ways. It is both a repetition and an intentional conscious choice of what could be useful as a shortcut, in order to save time.

I have learned all the factors of any number up to 1000 (including 301=7x43) and I described the whole process in my last post of this thread https://artofmemory.com/forums/prime-numbers-mental-calculation-and-memorization-2749.html (in my post on Nov-9-2015)

Regards,
Nodas

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16

Before my latest distraction… Got an interesting job, my 3x3 was coming along nicely. There is something to be said for daily progressive practice and a focus on the basics. In many ways technique is less important than perserverance.

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17

Do you mean all the factors, or all the prime factors? It seems like you were only talking about prime factors in that post you linked, and that seems like it would be the most helpful.

18

Just the primes. The composites are easier, once you master the primes, you can see instantly if a small number up to 999 is prime or not.

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