# Mental Calculation



## cmhardw (Oct 5, 2008)

Hi everyone,

I tried my first ever attempt at multiplying two 8-digit numbers mentally today. I got the answer incorrect (I was off by one digit), and it took me 22:22.25 for the attempt which I timed on Netcube.

I use a technique that I came up with on my own, and I think I am interested in practicing this more. I have also read up on how to square root mentally, and though I have never timed it I have practiced taking square roots of 6 digit numbers out to one decimal place mentally.

Does anyone else here do mental calculation?

Chris

P.S. The two numbers I was multiplying I generated here: www.random.org and they were 45483670 * 94052777 = 4277 865471 651590. My answer was 4277 765471 651590. Hey my answer was only off by one hundred billion, that can't be too bad right? ;-)

P.P.S yes I did search the forum before posting this thread, so you search nazis can calm down.

--edit--
Second attempt, Oh my god I got one! This actually isn't as hard as it sounds. When I first heard of the World Cup for Mental Calculation, and that this was one of the events, I thought "dang that's ridiculous." But it really isn't that hard to be perfectly honest, it feels very similar to blindsolving actually.

Ok so second attempt, again using random.org to generate my numbers:
56339606 * 53274048 = 3001438874345088

Got it correct in 16:07.13 timed on Netcube. Wow this is kinda fun!
--edit--


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## pjk (Oct 5, 2008)

Nice work Chris. I used to practice multiplying 3 digit numbers mentally back when I was in 7th grade or so. I haven't tried since, but I think I may start now that you bring this up.

Any specific techniques you're using? Can you run through the process that your mind went through while doing your 2 8-digit calculations today?


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## hawkmp4 (Oct 5, 2008)

That's amazing! I'm much better with mental math that I can visualize (calculus is a piece of cake for me).
Operations I'm not so good at. I'd be very interested in learning to take roots mentally, I've looked around casually at some sites but maybe now I'll actually do it


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## ThePizzaGuy92 (Oct 5, 2008)

wow how interesting, i've never heard of this, i'll definitely look this up


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## cmhardw (Oct 5, 2008)

pjk said:


> Nice work Chris. I used to practice multiplying 3 digit numbers mentally back when I was in 7th grade or so. I haven't tried since, but I think I may start now that you bring this up.
> 
> Any specific techniques you're using? Can you run through the process that your mind went through while doing your 2 8-digit calculations today?



Hey Patrick,

Haha I didn't start doing mental calculation until about 2 years ago, when I started my tutoring job. That's crazy that you were doing 3 digit times 3 digit in 7th grade, much respect.

As to the method I use, I did come up with this on my own. However, seeing as it's pretty much only based on the distributive property I think it's probably been discovered by hundreds of other people as well.

Say I wanted to do 845 * 241

I would see this in my mind's eye (the underscores are there only to keep the formatting the way I want it to appear):

__|845|
14|2 |

2*8 is 16 so I remember "16".

_|845|
1|42 |

At this step I do 4*8+2*4. When the numbers are written the regular way I view it sort of as a criss-crossing multiplication. For this step you get 40.

Now take my 16 from before and add the 40 like this:
16
_40
----
200

Then I say or think "200" Next step:

|845|
|142|

Here I do 1*8+4*4+2*5 = 34

Add it to the 200 like this
200
__34
-----
2034

Next step:
|845|
|_14|2

Here I do 1*4+4*5 = 24

Add it like this:
2034
___24
------
20364

Next step:
|845|
|__1|42

Do 1*5 = 5

Add it like this:
20364
_____5
-------
203645 for your final answer.

I basically did the same thing for the 8-digit times 8-digit, only I wrote down 1 or 2 digits at a time for my answer when I was sure they would be part of the final answer. This way I wouldn't have to try to remember them for the rest of the calculation. I don't know if this is allowed by the rules for the Mental Calculation World Cup, I am just trying it for the first time lol.

Chris


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## F.P. (Oct 5, 2008)

Yep, I do mental calculation too. 

Do you know Rüdiger Gamm?


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## Mozza314 (Oct 5, 2008)

I remember doing 4 digit by 1 digit problems in my head back in about 3rd grade 

A few years ago I saw a show about the savant Daniel Tammet, who was asked to do 37^4 in his head, doing it in about 20 seconds. I was amazed and decided to try out the calculation myself, I forget how long it took but it was several minutes. I've since done, I think a 2 digit number to the power 6, but I haven't been interested enough to practice.


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## blah (Oct 5, 2008)

I thought Bernett Orlando was pretty good at this stuff? Maybe you could email him or his dad or something?


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## fanwuq (Oct 5, 2008)

I remember back in elementary school, I heard of many crazy Chinese people who can do this very fast. I had a booklet about some methods, maybe I can try to find it...
Sounds very interesting. Good work, Chris!


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## PatrickJameson (Oct 5, 2008)

Nice work! I do mental calculation for fun during school sometimes. I don't usually go past 5 digits though  I use a "cross" method. I don't exactly know the name for it. 

ex.
429
354

(9*4)=first digit from the right(Answer 36; I carry the 3, in this case, to the next step; remember 6)

3+(2*4)+(5*9)=second(Answer 56; Carry the 5; remember 66)

5+(4*4)+(9*3)+(2*5)=3rd(Answer 58; Carry the 5; remember 866)

5+(4*5)+(3*2)=4th(Answer 31; Carry the 3; remember 1866)

(4*3)=5th(answer 15; answer to problem 151866)

151866 is the answer

Now that you bring this up I may try for higher numbers.


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## EmersonHerrmann (Oct 5, 2008)

I do short mental calculations sometimes...I'm gonna try your method out Chris, thanks for posting a thread on this! (Hey that rhymes xD)


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## sheriff (Oct 5, 2008)

can you write digit by digit when you know they'll be in the final answer?


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## MistArts (Oct 5, 2008)

sheriff said:


> can you write digit by digit when you know they'll be in the final answer?



That ruins the mental part.


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## cmhardw (Oct 5, 2008)

sheriff said:


> can you write digit by digit when you know they'll be in the final answer?



I found this from memoriad.com:



> (There will be two attempts at this category with different sets of questions each time.)
> 
> Time: 15 Minutes
> 
> ...



The wording seems confusing to me, but I'm thinking it is ok to do what I did, which was write 1 or 2 digits of the final answer, then calculate, then write 1 or 2 more digits, then calculate, etc..

I don't know if that is stretching the rules though. Basically you are still writing only the final answer, but v....e.....r....y........s.....l.....o....w.....l.....y......

Chris


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## PatrickJameson (Oct 5, 2008)

> As the competitor makes the mental multiplication, he can write the last result of the multiplication from right to left or from left to right in order.



I think this means you can write the answer but nothing else.


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## Brett (Oct 5, 2008)

When I get bored in class I sometimes will write a 3 digit and 4 digit number and multiply them. (mentally)

Those are the largest I can do :/ I may try higher order numbers if I get extremely bored...


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## cmhardw (Oct 5, 2008)

PatrickJameson said:


> I think this means you can write the answer but nothing else.



I was only confused on whether I can take 16 minutes to write my answer down, 1 or 2 digits at a time, or if I must write it down in 2-3 seconds at the end.

Chris


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## qqwref (Oct 5, 2008)

I used to do some mental multiplication. I never tried to do it all with my memory, but I can still multiply numbers on paper without writing down anything but the answer  I think the fastest I got was around two seconds per pair of digits multiplied, which would work out to a bit over two minutes for an 8-digit multiplication. I tried the memoriad example one just now and got 4:30 but was off by 2900 (three digits wrong).


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## NoahE (Oct 5, 2008)

PatrickJameson said:


> Nice work! I do mental calculation for fun during school sometimes. I don't usually go past 5 digits though  I use a "cross" method. I don't exactly know the name for it.
> 
> ex.
> 429
> ...




how would you do it for 4 or 5 numbers though


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## PatrickJameson (Oct 6, 2008)

NoahE said:


> PatrickJameson said:
> 
> 
> > Nice work! I do mental calculation for fun during school sometimes. I don't usually go past 5 digits though  I use a "cross" method. I don't exactly know the name for it.
> ...



http://img139.imageshack.us/img139/2830/mentalmathxc0.jpg
4 digits^^

http://img396.imageshack.us/img396/2406/math5kf3.jpg
5 digits^^

You should be able to figure out 5+ digits pretty easily.

Edit: I tried an 8 digit x 8 digit and got 6:59.69. I was off by 3 digits though


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## fanwuq (Oct 6, 2008)

PatrickJameson said:


> NoahE said:
> 
> 
> > PatrickJameson said:
> ...



Nice method! I'm going try it using your method.


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## F.P. (Oct 6, 2008)

Just get "Dead Reckoning: Calculating Without Instruments" (Ronald E. Doerfler)...best one around there.


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## choipster (Oct 10, 2008)

are you allowed to write down the two numbers you're multiplying? I just tried with 241 and 365 and I kept forgetting what numbers I was supposed to be multiplying >_>;;

either way, it would probably take some time for me. I don't really have a method per se. I just visual solving it on a piece of paper in my mind.


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## Hepheron (Oct 10, 2008)

Ive been expirimenting with mental calculations as well. When you are performing these feats do you need to memorizr the number or can you look at the problem?


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## Hepheron (Oct 10, 2008)

the abacus http://www.youtube.com/watch?v=DbiTEzBk8R8


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## fanwuq (Oct 10, 2008)

Hepheron said:


> the abacus http://www.youtube.com/watch?v=DbiTEzBk8R8



Wow, that's amazing!

I had to use this in elementary school, but I forget how it works. 
It's pretty fast, faster than a calculator sometimes.


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## cookingfat (Oct 10, 2008)

that abacus thing is insane. 

I also tried out that 'cross' method to multiply two 3-digit numbers and it's great, I could never do anything like this before.


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## ThePizzaGuy92 (Oct 10, 2008)

wow, that "cross" method is so cool!! this is on it's way to being one of my hobbies, haha, all because of this thread


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## cmhardw (Jul 20, 2010)

Thread bump. I've started practicing mental calculation again recently, and wanted to see if others were interested in it as well.

I've been using this site, and have been focusing mainly on 3 digit times 3 digit multiplication recently. I'm down to about 35 seconds average per problem, with maybe an 80%-90% accuracy rate. Mental calculation is quite a lot of fun, and I still compare it to BLD solving. I use a technique I "discovered" on my own, and I calculate left to right for multiplication. I'm working on writing down my answer as I go so I don't have to remember a lot of digits at a time, but there is a bit of an art to knowing when a portion of your answer is definitely part of the final answer.

Hopefully there are some other mental calculators out there, as this is getting kind of fun! I know Bernett is quite good at calculation, and I'm sure others who posted in this thread the first time around are still interested.

Chris


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## Ranzha (Jul 21, 2010)

Wow, good thread to bump.

I've been trying to work on squaring 3-digit numbers completely mentally.
Some people don't understand my method as to doing this, but you guys might.

Try this on paper, first. It helps a LOT.

Okay, say the number is 139, and you must mentally square it.
Already knowing after about 35 seconds that the answer is 19,321, I'll show you how I got it.

Okay. So I split the number 139 into two parts. You can do this in two ways:
13|9 or 1|39.
Square the figures on either side of the split.
I'd personally use the first split.
13^2 = 169; 9^2 = 81.
Form these two results as one five-or-six-digit number.
In this case, I'd join 169 to 81 to form 16981.
Remember 16981! Or write it down for reference.

Now we take the figures of the split (13 and 9) and multiply them together.
13*9 = 117.
Now multiply that by two.
117*2 = 234.
Since the split occurred after the tens' place, add one zero at the end of 234 to make it 2340.
(Similarly, if the split was after the hundreds' place, you would add two zeroes.)

Add 2340 to 16981.
19,321.


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## dillonbladez (Jul 21, 2010)

I used to do abacus too 

I could do 4x3 digit multiplication too. I went to Abacus Brain Study, i'm pretty sure it's not well known, but there are some classes in the U.S.

I didn't have fond memories, but it is useful.


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## incessantcheese (Jul 21, 2010)

Ranzha V. Emodrach said:


> Wow, good thread to bump.
> 
> I've been trying to work on squaring 3-digit numbers completely mentally.
> Some people don't understand my method as to doing this, but you guys might.



i think this is how most people square in their heads. (a+b)^2 expanded. except i guess you drop and add 0s in your head instead of keeping it as part of the calculations.


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## cmhardw (Jul 21, 2010)

Ranzha V. Emodrach said:


> I've been trying to work on squaring 3-digit numbers completely mentally.



Hey Ranzha,

This is interesting. I like your approach for squaring, and I've thought about this, squaring 3 digit numbers, a bit myself. I have my single and double digit squares pretty well memorized, if not by reflex at least with a small memory delay, from 1 to 61. This is part of a long term goal of knowing them from 1 to 99.

I have debated with using a special approach to squaring 3 digit numbers compared to just my regular method for multiplying two 3 digit numbers.

For example, to square 139 I would just use my normal multiplication method - the only exception being that I can use my knowledge of squares a little bit as follows.

139
139

I would split the numbers up as:

1|39
1|39

and I multiply left to right.

First step: 1x1=1 so I think 1

next step is a sort of criss cross of 1x39 (lower left 1 times upper right 39) + 39x1 (lower right 39 times upper left 1). This is 78. Because I shifted back 2 place values to move from the 1 to the 9 in 39 I have to add this to the 1 I already used, but to ensure that I shift right by 2 place values first. So I get this:

1
_78
----
178

Ok next step is to multiply both 39's together, but I already have memorized that this is 1521. I had to shift back two place values right to get from the 1 to the 9 in 39, so I have to shift my answer right two place values to ensure I add correctly.

So I add
178
_1521
------
19321

What do you think?

Chris


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## JeffDelucia (Jul 21, 2010)

This is the best off topic thread I have ever read. I have always loved math and I have always been naturally good at mental calculation but not until now have I thought of actually practicing it. Using Rahnza's method I can now square up to 3 digit numbers in my head in about 30 seconds.


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## Ranzha (Jul 21, 2010)

Spoiler






cmhardw said:


> Ranzha V. Emodrach said:
> 
> 
> > I've been trying to work on squaring 3-digit numbers completely mentally.
> ...






Hm. This is almost like my method, only with the variation of two of the steps.
I find this method you described useful for two- and three-digit numbers, but I just found a way that these methods can be adapted to square, say, a five-digit number.



Spoiler



37586 squared.
1,412,707,396 is the answer.
Here's how you get it.

Okay, as I said before, split the number into two more easily squarable chunks.
In this case, I'll do 375|86. (This is because having a number to be squared ending in 5 is a wonderful thing.)

Okay. 375 squared, using ending-in-five method:
Okay, 37(0)^2 is 1369(00).
37*5*2(*10) = 37*10(*10) = 370(0).
136,900 + 3700 = 140,600.
5^2 = 25.
140600 + 25 = 140,625.

Then! 86 squared:
6,400 + 960 + 36 = 7,396.

Now then!
140,625 * 100^2 = 1,406,250,000.
Add 7,396 = 1,406,257,396.

Then, multiply 375(00) by 86 and multiply by two = 6,450,000.

Add 1,406,257,396 and 6,450,000:
1,412,707,396.



It's not a good mental math method, but it works for sure.

--Ranzha


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## Ranzha (Jul 21, 2010)

JeffDelucia said:


> This is the best off topic thread I have ever read. I have always loved math and I have always been naturally good at mental calculation but not until now have I thought of actually practicing it. Using Rahnza's method I can now square up to 3 digit numbers in my head in about 30 seconds.



Sorry for double post (I got ninja'd), but why thank you, Jeff!
It always brings joy to me when people understand the stuff that goes on in my head. xD

anyway, I call this method "divide and conquer." xD


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## blakedacuber (Jul 21, 2010)

Once when i was in bed a few years ago(im 16 now)i think i was 12:/ but anyway i figured out that squares have a patrn and tryed to explain this to several math teachers non oof which understood how it worked
basiclly 2^2 is 4 yes?
3^2 is 9
so *9-4=5*
lets continue 
4^2 is 16
*16-9=7*
5^2 is 25
*25-16=9*
so basiclly the higher up in squares you go you go to the next odd number
I know its not as effficient as your methods but it works but just alot slower


Spoiler



ou last odd number was 9 so the next is 11 so if you ad this to 25 (5^2) you get 36 which would be the next square


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## Ranzha (Jul 21, 2010)

blakedacuber said:


> Once when i was in bed a few years ago(im 16 now)i think i was 12:/ but anyway i figured out that squares have a patrn and tryed to explain this to several math teachers non oof which understood how it worked
> basiclly 2^2 is 4 yes?
> 3^2 is 9
> so *9-4=5*
> ...



Did you notice that 3*3 - 2*2 = 3+2?

Additionally, this can be furthered.
Take two either both even or both odd numbers and multiply them.
Say these numbers are 27 and 35.
Now, I've found a cool way to get the answer (645) without just multiplying it out.

Find the average of the two numbers.
(27+35)/2 = 31.
Square it. 31^2 = 661.
Subtract 27 from 31 (or, alternatively, 31 from 35). 31-27 = 4.
Square 4. 4^2 = 16.
661-16 = 645.
645 is the answer.

The product of any two numbers x and y is:
((x+y)/2)^2 - ((x-y)/2)^2.

Let's try 4 and 5 as x and y.

4*5 =
((4+5)/2)^2 - ((4-5)/2)^2
(9/2)^2 - (-1/2)^2
4.5^2 - -0.5^2
20.25 - 0.25
20.

--Ranzha


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## jms_gears1 (Jul 21, 2010)

Ranzha V. Emodrach said:


> JeffDelucia said:
> 
> 
> > This is the best off topic thread I have ever read. I have always loved math and I have always been naturally good at mental calculation but not until now have I thought of actually practicing it. Using Rahnza's method I can now square up to 3 digit numbers in my head in about 30 seconds.
> ...


ranzha for the method name <3
imma start messing around with MentalCalcs and what not. Starttinnnnggggg thursday.


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## ThatGuy (Jul 21, 2010)

Challenge. How about finding the GCD of two insanely large numbers.


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## blakedacuber (Jul 21, 2010)

Greatest Common Dinominator?


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## ThatGuy (Jul 21, 2010)

Greatest Common Divisor.


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## cmhardw (Jul 21, 2010)

ThatGuy said:


> Challenge. How about finding the GCD of two insanely large numbers.



I don't know of a way to do this very quickly (i.e. shortcut). I would probably prime factorize the number and calculate it that way.

Btw, there is a very simply way to *construct* a divisibility rule for any prime number (except 2 and 5).

For example, I would like a rule to know when a number is divisible by 17. To do this I need to multiply 17 by a number until the result ends in a nine. I will multiply 17 by 7 to do this, and get 119. Now I truncate the ones place from 119 to make it 11, and lastly add 1 to this to get 12. 12 is a number very closely related to 17 in terms of divisibility. The method I learned called 12 the "X-factor" of 17.

Ok, so now that we have the X-factor of 17, we have a very simple divisibility rule for 17. I will use 17^3 = 4913 to show this. We know that 4913 is divisible by 17, but let's use the X-factor for this.

1) First truncate the ones digit off the 4913. This gives us:

491 | 3

Multiply the truncated 3 by 17's X-factor. 3*12=36. Now add this result to the remaining 491 that is left over.

491
_36
----
527

If this result is divisible by 17, so is our original number. 527 is divisible by 17, but you can iterate the process if you do this mentally.

52 | 7
7*12 = 84

52
84
--
136 (divisible by 17, and therefore our original number is as well)

13 | 6
6*12 = 72

13
72
--
85 (divisible by 17, and therefore our original number is as well).

----------------------------------------

There is one other useful thing to know about doing this. 12 is the X-factor of 17, and 17-12 = 5 is the complement X-factor.

So I could have done this much easier with the complement X-factor.

491 | 3
3*5 = 15

And if you use the complement X-factor you subtract the result.

491
_15
----
476

47 | 6
6*5 = 30

47
30
---
17 (and so 4913 is divisible by 17)

---------------------------

I also find it fun to prime factorize larger numbers using the X-factors of all the primes less than or equal to its square root. You could then use the prime factorizations of each of the two numbers to calculate the greatest common divisor.

Chris


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## ThatGuy (Jul 21, 2010)

Spoiler






cmhardw said:


> ThatGuy said:
> 
> 
> > Challenge. How about finding the GCD of two insanely large numbers.
> ...






Let's start with two smaller number, lets say 17 and 7.
17=(2)*7*+3
*7*=(3)*2*+1
*2*= (1)2+0
Once you reach 0 look at the previous line and find the remainder, which in this case is 1. That number is the GCD of those two numbers.

24 16
24=(1)16+8
16=(2)8+0
So 8 is the GCD.
This can be proven using modular arithmetic.
http://en.wikipedia.org/wiki/Euclidean_algorithm

So, if we take two large numbers we can apply many different mental calculations as well as memory.


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## cmhardw (Jul 21, 2010)

ThatGuy said:


> Spoiler
> 
> 
> 
> ...



:fp Euclidean algorithm, nice... Should have thought of that lol.

But you do have to admit, that X-factor stuff is pretty cool. I always hated it when my teachers used to say that the divisibility rule for 7 was "so incredibly difficult that it will melt your mind into a goo, and that goo will then have to be scraped off the classroom floor" or something to that effect 

7 has a complement X-factor of 2, so following the same process as above you have a relatively easy divisibility rule for 7.

Chris


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## ThatGuy (Jul 21, 2010)

I remember in 6th grade the teacher gave us a wrong divisibility by 7 so I ended up thinking that none existed. I didn't really do much thinking about it until I went to Ross and then I was like OMG so this is where so much of the random stuff they tell us comes from.


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## Sakarie (Jul 21, 2010)

Ranzha V. Emodrach said:


> Wow, good thread to bump.
> 
> I've been trying to work on squaring 3-digit numbers completely mentally.
> Some people don't understand my method as to doing this, but you guys might.
> ...



It's a fascinating method, but it's not perfect. Take 328, and split 3|28, and see what result you get.


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## Ranzha (Jul 21, 2010)

Sakarie said:


> Ranzha V. Emodrach said:
> 
> 
> > Wow, good thread to bump.
> ...



Oh, I forgot to explain the other split. I assumed one would get it, but I guess not.

Okay, since the split is at 3|28, you square three and twenty-eight.
3^2 = 9; 28^2 = 784.
Since three was in the 100s' place, you're actually squaring 300, and thus the five digit number is 90784 (90000 + 784).

Since three from 328 is in the hundreds' place (as I stated before), you multiply 300 by 28, and multiply by two, since in the method before, we added the zeroes at the end.
300 * 28 * 2 = 16800.

90,784 + 16,800 = 107,584.
And that is the answer.


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## Carrot (Jul 21, 2010)

wouw... am I supposed to read all this? x'D 

just to get on topic, I have tried some Mental Calculations a bit, but I calculate from right to left..

I have worked out my own method for both multiplying numbers and taking the third root of a number, if it's answer is a 2 digit number (very easy )

How I multiply in head(Yes, it sucks =D ):


Spoiler



pretty much a criss cross one... but it works 

let's take a number: abcd * efgh = jklmnop

the numbers are just spots...

p=h*d
o=g*d+c*h+truncated(p/10)
n=b*h+c*g+d*f+truncated(o/10)
m=a*h+b*g+c*f+d*e+truncated(n/10)
l=a*g+b*f+c*e+truncated(m/10)
k=a*f+b*e+truncated(l/10)
j=a*e+truncated(k/10)

yeah =D


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## cmhardw (Jul 22, 2010)

I just tried two attempts at 8 digit times 8 digit mental multiplication. The only thing I wrote down was the answer, no intermediate work. I used www.random.org to generate the numbers, and timed the attempts with netcube.

First attempt was off by quite a lot:
94819663 * 54799651 = 5196 084440 337613

I got the answer wrong with 5295 984340 337613. This attempt took 21:03.95 minutes.
---------

Second attempt was:
64602573 * 90381014 = 5838 846054 749022

and I got the answer correct for this one in 9:24.85 minutes. I have no idea why this attempt was so much faster. Probably due to lack of practice (I'm still on the steep part of the learning curve), as well as having more 0's in the two numbers on the second attempt. I'm glad for my first Stackmat time at doing this at least, but I want to wait until I do it again before I get excited.

Chris


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## qqwref (Jul 22, 2010)

Did you allow yourself to write down digits of the answer before you got the whole thing, or did you try to keep the entire thing in memory? I think I could do the multiplication in a lot under 10 minutes without writing down anything *but the answer*, but I wouldn't be able to do it without writing down anything *until I know the answer*.


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## cmhardw (Jul 23, 2010)

qqwref said:


> Did you allow yourself to write down digits of the answer before you got the whole thing, or did you try to keep the entire thing in memory? I think I could do the multiplication in a lot under 10 minutes without writing down anything *but the answer*, but I wouldn't be able to do it without writing down anything *until I know the answer*.



I don't know the official rules for this, and it's something I know has discussed in the earlier posts in this same thread from a long time ago. I wrote down the digits once I was sure they were part of the final answer, but I was not done calculating as I wrote them down. So, I did technically write down only the answer, and no intermediate calculations. However, I took the entire 10 minutes, or 20 minutes, to do so.

I would rather practice this by following the official rules for mental calculation competitions, but I haven't researched enough to know what those rules are. I'll try to take a look into this and see what those who specialize in this say about it, but I hope you are allowed to write down digits without knowing the entire answer already.

Chris


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## cmhardw (Jul 23, 2010)

I found this page of records for calculating which lists some rules for how the attempts were done.

Some relevant rules are:


> 5. The calculator should write down the answer to the calculation. If the answer is written down, this can be done left-to-right, right-to-left or in any arbitrary order.
> 
> 6. The timing begins when the two numbers become visible to the competitor and ends at the end of writing the answer.
> 
> 7. In some cases, the calculator may dictate the answer - then the timing ends as the calculator finishes dictating the answer.



So it seems it's similar to our blindfold solving. The total time is from when you see the numbers to when the answer is written down. I would assume from this that you should be able to write down the numbers as you calculate, and that you don't have to write it only once you know the entire answer.

Chris


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## The Bloody Talon (Jul 23, 2010)

when im in highschool, i used to challenge my classmates in square rooting a 5 or 6 digit perfect square mentally with a given number from a calculator
(random number x 1000)^2
anyway, my job involves mathematics, but I think I'm gonna stick with the calculators... 
but if I gonna have free time, maybe i'm gonna try that


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## ThatGuy (Jul 23, 2010)

I tried 111x158 while I was doing nothing. I didn't really count time but I got it right  It was sort of brute force in my head so when I tried xyz with x>1 I failed...but it was a good time passer.


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## Mike Hughey (Jul 23, 2010)

cmhardw said:


> I found this page of records for calculating which lists some rules for how the attempts were done.
> 
> Some relevant rules are:
> 
> ...



I've read these before, and I thought I remembered that being the way it worked. Also, the rules for the World Cup here follow the same general guidelines.


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## ssb150388 (Jul 23, 2010)

blakedacuber said:


> Once when i was in bed a few years ago(im 16 now)i think i was 12:/ but anyway i figured out that squares have a patrn and tryed to explain this to several math teachers non oof which understood how it worked
> basiclly 2^2 is 4 yes?
> 3^2 is 9
> so *9-4=5*
> ...



So basically, if you are trying to calculate N^2, it is addition of first N odd numbers. Nth odd number is 2*N-1. So if you want to add the numbers, you will get N/2 pairs, each adding up to 2N. So basically you are doing, 2N * N/2 which is N^2.

So if you go in the reverse order, you have a proof for your theory.


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## Ranzha (Jul 23, 2010)

ssb150388 said:


> blakedacuber said:
> 
> 
> > Once when i was in bed a few years ago(im 16 now)i think i was 12:/ but anyway i figured out that squares have a patrn and tryed to explain this to several math teachers non oof which understood how it worked
> ...



The way I saw this was that the difference between two numbers' squares is the sum of the two original numbers if the two original numbers' difference is one.
To prove:

5*5 = 25.
6*6 = 36.
36-25 = 11.
5+6 = 11.

5*5+5 = 5(5) + 5(1) = 5*6.
5*6+6 = 6(5) + 6(1) = 6*6.
Therefore, 5*5+5+6 = 6*6.

To further prove:
2.4*2.4 = 5.76.
3.4*3.4 = 11.56.
11.56-5.76 = 5.8.
2.4+3.4 = 5.8.

2.4*2.4+2.4 = 2.4(2.4) + 2.4(1) = 24*3.4.
2.4*3.4+3.4 = 3.4(2.4) + 3.4(1) = 3.4*3.4.
Therefore, 2.4*2.4+2.4+3.4 = 3.4*3.4.

Additionally, the difference of the squares of two numbers whose difference is two is two times the sum of the two numbers given.
Basically, if y-x = 2, then y^2 - x^2 = 2(x+y).


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## cmhardw (Oct 2, 2012)

I've been working on mental calculation recently again and wanted to post about it. This thread already exists, so I figure it's better to bump here than start a new thread.

I recently finished learning all my squares from 1 to 100 by memory. Some of the squares I still have to calculate quickly on the spot (larger squares that end in 5 for instance, like 65^2, 75^2, etc.), but most of them I have memorized and can recall the answer within 3-4 seconds after using various memory mnemonics.

I'm using these to be able to help with mental multiplication in general, but I am also practicing how to square 3 digit and 4 digit numbers in my head also, using something very similar to the regular multiplication of numbers method that I use for numbers 2 digits and larger, but using the memorized squares up to 99^2 to make this process much simpler and much quicker.

I know there are others on this forum who practice mental math, as evidenced by the replies in this thread from when I first posted it. I'm sort of learning things for fun without much of a solid goal right now. I guess I am loosely interested in one day maybe participating in the mental calculation world cup, but I am not really practicing for their events with any serious effort yet.

Next I plan to expand my multiplication tables from 12x12 to 25x25 from memory. I already have the squares up to 25x25 memorized as mentioned, and I'm not sure yet how I'll work on the rest. I think I'll work up to 13x13, then 14x14, then 15x15, etc but I'm not sure yet.

Not sure if others will find this interesting, but I am excited about my progress so far and wanted to post about it.


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## peterbone (Oct 3, 2012)

I used to multiply numbers. I could do two 4 digit numbers fairly easily (maybe around 30 second) and sometimes two 5 digit numbers. My method limited me to no more than 5 since that's the amount of constantly changing numbers I could keep track of.


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## Ranzha (Oct 5, 2012)

After a couple years, rereading your method for squaring 3-digits, that's so elegant. Same steps, just different order. Me gusta.

My usual approach to multiplying integers is to prime factorise until I have easier numbers to work with. I usually apply for up to 3-digit numbers.


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## ben1996123 (Oct 6, 2012)

I would try this, but I probably wouldn't be very good at it, since I just tried to mentally calculate 23-16 and it took me 10 seconds. If I write it down though, I can do it straight away... I've always been really bad at basic addition and subtraction with numbers above 10. How iroñic, since I find all calculus that I've done so far really easy.


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## Ickathu (Oct 8, 2012)

Teacher: "Who know's what 1+1 is?"
Ben: *raises hand*
Teacher: "yes ben?"
Ben: "11!"

*later*
T: "What's the limit of d(sin^-1(x))*e^cos(x)?"
B: *raises hand*
T: *sighs* "Yes, Ben?"
B: "[insert completely correct answer + information about how the problem should be worded properly + new found mathematical theory]"


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## 5BLD (Oct 8, 2012)

I am good with complicated stuff like weird diffing, but big numbers scare me

O and i can square 2-3 digit numbs quite fast. 2 digits take just a few seconds and I haven't learnt the majority of them.


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## ben1996123 (Oct 9, 2012)

Ickathu said:


> Teacher: "Who know's what 1+1 is?"
> Ben: *raises hand*
> Teacher: "yes ben?"
> Ben: "11!"
> ...



yeah cuz 1 is a number above 10, rye?


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## cmhardw (Jan 9, 2014)

*Squaring numbers mentally - update*

Hey everyone! I felt like it was time for an update 

I'm still practicing mental calculation, specifically squaring. I have the squares of all the two digit numbers memorized, or at least I have a specific process to calculate the square very quickly for numbers of a certain category which I use for numbers in the 40s, 50s and 90s. These are slowly sinking into long term memory as well, but they are not all there yet.

I am fairly comfortable squaring 3 and 4 digit numbers mentally now, and I am getting faster at them the more I practice. I haven't done any timed trials in a while, but I may do so just to guage where I am at. I have been working for the past couple months on squaring 5 digit numbers, and I have a process that I am now fairly comfortable with. Starting today I began working on having a special case approach for numbers of the form:
(_ _ 0 _ _)^2 as well as
(_ _ 5 _ _)^2

I have the approach for the first type memorized, but I am still working on the second.

My medium term goal is to work up to squaring 6 digit numbers, then 7, then 8. After I am comfortable squaring 8 digit numbers mentally I would like to start really practicing multiplying two arbitrary 8 digit numbers mentally. I already see one way that I can do this more efficiently than I have tried in the past. In the past I was working one digit at a time, but after working with squaring 4 and 5 digit numbers I have seen the advantage of being able to work with double digit numbers for all intermediate calculations. Doing this for multiplying any arbitrary 8 digit numbers would greatly speed things up I think, but I may also consider splitting up the 8 digit number problems into groups of 4 digits as well.

Anyway, that's my update! I still practice this daily, though I by no means am training it by any serious competitive standard. My philosophy is very much in line with the saying "It is better to light a candle than to curse the darkness"

I hope those who have posted earlier in this thread are still interested in mental calculation, and if so I would be interested to hear your progress!


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## kcl (Jan 11, 2014)

Chris, that's awesome 
I should start doing that.


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## PeelingStickers (Jan 11, 2014)

Dang I only know up to 32^2 memorized and 32^3, I should memorize more


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## A Leman (Jan 13, 2014)

cmhardw said:


> Hey everyone! I felt like it was time for an update
> 
> I'm still practicing mental calculation, specifically squaring. I have the squares of all the two digit numbers memorized, or at least I have a specific process to calculate the square very quickly for numbers of a certain category which I use for numbers in the 40s, 50s and 90s. These are slowly sinking into long term memory as well, but they are not all there yet.
> 
> ...



That's great! I was interested in mental math before cubing, but I have not practiced seriously since cubing. I was trying the problems on MEMORAID recently which are 8 digit x 8 digit multiplication! I was using my image list to keep track of the numbers. Do you do the same? It's not important for smaller problems, but I would have been completely lost if I was not using letter pairs. 

I also saw this earlier in the thread:


cmhardw said:


> Some of the squares I still have to calculate quickly on the spot (larger squares that end in 5 for instance, like 65^2, 75^2, etc.)



I don't know what you use for squares. I normally use the difference of 2 squares rule from algebra to make them easier problems. For example, 48^2= 
46*50+2^2 = 2304 

when you do this by going up five and down five for numbers ending with five you always get tailing zeroes in the multiplication stage and 5^2 is 25 so the last two digits are always 25.

What this means for me is that 65^2= 6*7 and add a 25 at the end so 4225 and 
75^2=7*8=56...25

so 645^2=64*65...25=416025

It makes the squares that end in 5 freebies until the number in front of it is very difficult.


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## cmhardw (Jan 13, 2014)

Thanks to those who responded! It's great to meet others who also enjoy mental calculation! I don't know why I enjoy it, I just do  It's fun to know that others feel the same way!



A Leman said:


> That's great! I was interested in mental math before cubing, but I have not practiced seriously since cubing. I was trying the problems on MEMORAID recently which are 8 digit x 8 digit multiplication! I was using my image list to keep track of the numbers. Do you do the same? It's not important for smaller problems, but I would have been completely lost if I was not using letter pairs.



I didn't know that the 8 digit x 8 digit multiplication problems were from MEMORIAD, but I have heard of multiplication problems that large. I also read about square rooting 6 digit numbers being an event at some mental calculation competition(s) (MEMORIAD?). Sometimes I like to mentally square root my answer just to make sure I get the number I stared with. I've only tried this with 6 digit answers, and even then I don't do it often.

When squaring 5 digit numbers I do use mnemonics to help remember the answer. I use the MAJOR system, but as I get to larger and large multiplication problems I will probably need something more robust, like letter pairs, to remember the final answer. Are you allowed, in MEMORIAD, to write down the answer as you go? This would avoid having to mentally remember digits once you are _sure_ that those are the final digits in the answer.



A Leman said:


> I don't know what you use for squares. I normally use the difference of 2 squares rule from algebra to make them easier problems. For example, 48^2=
> 46*50+2^2 = 2304
> 
> when you do this by going up five and down five for numbers ending with five you always get tailing zeroes in the multiplication stage and 5^2 is 25 so the last two digits are always 25.
> ...



I do use the difference of squares when it is available/useful, yes. I also use the trick you mention where a number (10n+5)^2=100*(n)*(n+1)+25 with n a natural number.

I don't really use this trick often for 3 digit numbers that end in 5, as usually my normal squaring method is faster for me. Sometimes a three digit number that ends in 5 is clearly evident in the context of a larger problem, and in those cases I sometimes use the above trick for numbers ending in 5.

For 645^2 I would use my normal 3 digit method:

645^2
(6|45)^2
-------------
6*(2*45)
(6*9)*10=540

6^2 = 36

36
+540
-----
4140

45^2=2025

4140
+ 2025
--------
416025

My method keeps track of place value by "moves left" or "moves right" or "n spaces left of" or "n spaces right of". The addition problems I typed above are formatted exactly as I see them in my mind when doing a problem.

An example where I might use difference of squares is:
4019^2

(40|19)^2
40*2*19 = 40*38 = (39^2)-1 = 1521 -1 = 1520

40^2=1600

1600
+ 1520
-------
161520

19^2=361

161520
+____361
---------
16152361


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## A Leman (Jan 13, 2014)

cmhardw said:


> Thanks to those who responded! It's great to meet others who also enjoy mental calculation! I don't know why I enjoy it, I just do  It's fun to know that others feel the same way!
> 
> 
> 
> ...



Here is the link to the MEMORIAD software: http://www.memoriad.com/memoriadsoftware.asp

It also has the 6 digit square roots and since it is a computer software, you can not write down intermediate notes, just the digits for the answer as you go. It's a good way to practice 

I started using memoriad to practice mnemonics for numbers, but I realized that it's a bit ironic that I spent so much time in HS doing mental math and THOSE were the events that I am terrible at on memoriad. The memory stuff is still more fun than the mental math for me though.


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