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Submitted by Sav2718:

A part of a big list of cool number facts I’ve posted a few months ago (https://www.facebook.com/notes/gur-keren/cool-number-facts/590025287678630), many of which I discovered by myself and as far as I know I’m the only source of them. I chose this example (although can be generalized) because of its somewhat symmetrical nature. I call them “Train numbers”, “clone numbers” or “Gur numbers” (after myself ) .

Let’s have a look:
We claim that 166..6 (N 6s) times 4 is 66..64 (N 6s) (the exact same process of proof is true for our second example of 19,199,1999… times 5).

For N=1:
16*4=64 , true

Assume our claim is true for N=K-1 so lets have a look at N=K:

166..66 (K 6s) * 4=(166..66 [K-1 6s] * 10+6)*4=166..66 [K-1 6s]*10*4+6*4=(166..66 [K-1 6s]*4)*10+24 but we assumed our claim is true for K-1 6s so:
(166..66 [K-1 6s]*4)*10+24=(66..664 [K-1 6s])*10+24=66..6640+24 [K-1 6s]= 66..6664 [K 6s] so our assumption was correct.

Submitted by Sav2718:

I saw that someone wrote as a comment on my business card:
"I saw something similar for fifth roots- getting the last digit right is a lot easier too!".

So I just wanted to add a word about the rightmost digit of a perfect root in general:
The pattern for the rightmost digit is the same for every order of the form 4K+1 (the digit itself) and for 4K-1 (the same unique pattern as with cube roots). the same is true for roots of an even-order but an extra step of elimination is needed, two steps for even-order of the form 4K.

For example, lets take a look at some (perfect) square roots:

1^2=1
2^2=4
3^2=9
4^2=16
5^2=25
6^2=36
7^2=49
8^2=64
9^2=81
10^2=100
11^2=121
12^2=144
13^2=169
14^2=196
15^2=225
16^2=256
17^2=289
18^2=324
19^2=361
20^2=400
etc.

the pattern for the rightmost digit is always (0,)1,4,9,6,5,6,9,4,1(,0)
1 for 1 and 9 (the complement to 10)
4 for 2 and 8
9 for 3 and 7
6 for 4 and 6
5 for 5
a trail of 2N 0s (even number) in the right hand of the number beneath the root sign will be a trail of N 0s in right hand of the answer (ex: sqrt(15210000)=3900).

Now in order to calculate a given perfect square root lets look at an example:

sqrt(5329)

70^2(=4900)<5329<80^2(=6400)

or even more simplified:
7^2(=49)<53<8^2(=64)

so the leftmost digit of the answer is 7

5329 ends with a 9 so according to our pattern the rightmost digit is either 3 or 7 but how can we tell?

Let’s have a look at the trick for squaring a number that ends with 5:
(10a+5)^2=100a^2+2*10*5*a+5^2=100a^2+100a+25=100a*(a+1)+25

or even more simplified:

[a*(a+1)]&[25] (where “&” is the concatenation operator)

examples:
35^2=[3*4]&[25]=[12]&[25]=1225
75^2=[7*8]&[25]=[56]&[25]=5625
435^2=[43*44]&[25]=[1892]&[25]=189225

you may also prefer to calculate it as such:
35^2=[3^2+3]&[25]=[9+3]&[25]=[12]&[25]=1225
75^2=[7^2+7]&[25]=[49+7]&[25]=[56]&[25]=5625
435^2=[43^2+43]&[25]=[1849+43]&[25]=[1892]&[25]=189225

Now back to our square root:

we know sqrt(5329) is either 73 or 77 but we can quickly check that
75^2=[7*8]&[25]=[56]&[25]=5625>5329

so if 75 is too big, 77 is even bigger than the answer must be (correctly) 73.

Note that because a square of number ending with 5 always ends with 25 one doesn’t even need to calculate the whole thing and just have a look at 7*8 (7 times its successor).

Lets do another one quickly:

sqrt(107584)

32^2(=1024)<1075 (don’t need the ‘84’) <33^2(=1089)

32_

107584 ends with 4 so the rightmost digit is either 2 or 8

32^2+32=1024+32=1056<1075

too small so 322 is even smaller

so the answer is (correctly) 328.

As I already mentioned even-order roots from the form 4K need another step of elimination. 

Higher numbers require more steps and non-perfect roots require algorithms from a different kind and approach but I am willing to write about those in the future. 

He’s referring to this awesome submission.

Mental Arithmetic by Sav2718. This is a drawing inspired by mental calculation and synesthesia. Synesthesia is a neurological condition in which stimulation of one sensory or cognitive pathway leads to automatic, involuntary experiences in a second sensory or cognitive pathway. Some people who have synesthesia may perceive letters and numbers as inherently colored, while for some, days of the week and months of the year evoke personalities!

sav2718 said:

There is a big difference between loving numbers and loving math.A mental calculator named Willem Bouman (set the world record for factoring 5 digit numbers) said he hates math and consider himself an "arithmetician". That's why I chose to focus on number theory and algorithms, so I could express my love for both math and numbers. From my experience with high level education (I studied math and physics in Tel Aviv university) not only you don't need to know arithmetic, but professors s*k at it.

Somehow, giving a girl tips on multiplication tables has turned into a discussion about the difference between mathematics and arithmetic. This is just one of the many reasons why I love running this blog. Everyone has so much to say!

sav2718 said:

I do have synesthesia! I even got briefly mentioned in a final research project about synesthesia.

Wow! From Wikipedia:

Synesthesia is a neurological condition in which stimulation of one sensory or cognitive pathway leads to automatic, involuntary experiences in a second sensory or cognitive pathway. People who report such experiences are known as synesthetes. Recently, difficulties have been recognized in finding an adequate definition of synesthesia, as many different phenomena have been covered by this term and in many cases the term synesthesia (“union of senses”) seems to be a misnomer. A more accurate term for the phenomenon may be ideasthesia.
In one common form of synesthesia, known as grapheme (color synesthesia or color-graphemic synesthesia), letters or numbers are perceived as inherently colored, while in ordinal linguistic personification, numbers, days of the week and months of the year evoke personalities. In spatial-sequence, or number form synesthesia, numbers, months of the year, and/or days of the week elicit precise locations in space (for example, 1980 may be “farther away” than 1990), or may have a (three-dimensional) view of a year as a map (clockwise or counterclockwise). Yet another recently identified type, visual motion (sound synesthesia) involves hearing sounds in response to visual motion and flicker. Over 60 types of synesthesia have been reported, but only a fraction have been evaluated by scientific research. Even within one type, synesthetic perceptions vary in intensity and people vary in awareness of their synesthetic perceptions.

Watch on sav2718.tumblr.com

*!!!**sorry guys, I had to re-upload the video**!!!*

Myself, solving some “mate in one” chess puzzles.

Maybe I’ll do a “mate in two” version…..

A fun fact- I have a personification type synesthesia to chess pieces. 

Watch on sav2718.tumblr.com

Myself, demonstrating the limited-hold memory task to a higher level of difficulty and to a greater precision and accuracy than “Ayumu” the chimpanzee.

for more and other stuff:
facebook-https://www.facebook.com/pages/Gur-Keren/160006804013816
deviantart-http://sav2718.deviantart.com/

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