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flat assembler > Heap > A simple mathemagical trick for interested ones.

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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
An illusionist asks a randomly chosen spectator to pick (secretly) one of the natural numbers smaller than 100 and calculate its cube and give the result speaking it aloud. The illusionist immediately gives an answer, i.e. the original number chosen by the spectator. This trick doesn't require the magician to know all cubes of the numbers between 1 and 100. It's enough to remember only cubes of the first ten numbers from 1 to 10 because the method of determination of the original number is based solely on this information.

http://www.slideshare.net/fullscreen/mikhajduk/cube-root-49631334
Post 02 Jul 2015, 16:54
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Tyler



Joined: 19 Nov 2009
Posts: 1216
Location: NC, USA
Kaprekar's routine is another cool trick.
Post 06 Jul 2015, 17:56
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
Tyler wrote:
Kaprekar's routine is another cool trick.
Thanks for this link, I wasn't aware about existence of Kaprekar's constants before however I knew about the fact the following procedure:

1. Split the given (positive integer) number into separate digits.
2. Calculate the square of each digit.
3. Sum up all numbers calculated in the second step.
4. Go to the first step.

will lead us either to 1 or to the cycle

145, 42, 20, 4, 16, 37, 58, 89

M. Szurek, a Polish mathematician, wrote in his book that this kind of process had been called Steinhaus's problem although in Wikipedia numbers that eventually give 1 as a result of the aforementioned procedure are named "happy numbers".

D. R. Kaprekar was not only Ganit ("Mathematics" in Sanskrit) guru worth to mention here. I think that we cannot forget about Shakuntala Devi, who also wrote a few books on the theme of recreational mathematics.

Revisiting Numbers with Shakuntala Devi
Smile
Post 08 Jul 2015, 20:47
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shoorick



Joined: 25 Feb 2005
Posts: 1605
Location: Ukraine
he-he, as I'd studied mathematics only in the basic school, this simple trick looks for me like a real magic! the A upside down impressed me so much! maybe i'm not too expired yet to learn something beyond logarithms Cool

Mr. Szurik Smile
Post 10 Jul 2015, 11:17
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
shoorick wrote:
the A upside down impressed me so much!
The "A" symbol written upside down () is a "Western" counterpart for the symbol Λ ("wedge") used previously (before 90's) in Poland and other Central European and Eastern European countries as an abbreviation for the universal quantifier "for all". Wink

I remember how confused I was when I saw it for the first time in 1995 during math lectures at university. I don't know is the "new" notational convention truly better than the previous one but we all have to unify our math language to be better understood worldwide. Smile
Post 10 Jul 2015, 11:56
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shoorick



Joined: 25 Feb 2005
Posts: 1605
Location: Ukraine
now I see Smile to say true, if you were place there that lambda ("wedge"), i would not note the difference (but now i will, sure Smile )
Post 10 Jul 2015, 12:32
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Tyler



Joined: 19 Nov 2009
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Location: NC, USA
MHajduk wrote:
The "A" symbol written upside down () is a "Western" counterpart for the symbol Λ ("wedge") used previously (before 90's) in Poland and other Central European and Eastern European countries as an abbreviation for the universal quantifier "for all". Wink
Was the wedge not used for logical conjunction in Poland et al then? If it was, then using something new for "for all" does seem like an improvement. Of course, sometimes symbols have to be reused, but we should at least try to keep the different uses in mostly disjoint sub-disciplines. In this case, it would be double use of the same symbol in one fairly specific area, first-order logic, but at least the arity was different.
Post 12 Jul 2015, 06:33
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
Tyler wrote:
MHajduk wrote:
The "A" symbol written upside down () is a "Western" counterpart for the symbol Λ ("wedge") used previously (before 90's) in Poland and other Central European and Eastern European countries as an abbreviation for the universal quantifier "for all". Wink
Was the wedge not used for logical conjunction in Poland et al then? If it was, then using something new for "for all" does seem like an improvement. Of course, sometimes symbols have to be reused, but we should at least try to keep the different uses in mostly disjoint sub-disciplines. In this case, it would be double use of the same symbol in one fairly specific area, first-order logic, but at least the arity was different.
The symbols for universal and existential quantifiers were just enlarged versions for conjunction and alternative symbols respectively. It was quite logical because "for all" quantifier may be seen as a generalized conjunction for all elements of the given set whereas existential quantifier may be understood as a generalized alternative. Due to noticeable difference of sizes between "old-style" quantifiers and logical connectives there were no place for confusion.
Post 12 Jul 2015, 09:42
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Tyler



Joined: 19 Nov 2009
Posts: 1216
Location: NC, USA
MHajduk wrote:
It was quite logical because "for all" quantifier may be seen as a generalized conjunction for all elements of the given set whereas existential quantifier may be understood as a generalized alternative.
Oh yeah! That didn't occur to me. That's really nice. Kinda like big cup/cap are used to mean union/intersection of all sets in a collection. That's really elegant. We should change it back.
Post 14 Jul 2015, 23:21
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
Thank to Tyler who mentioned Kaprekar's routine, I've heard for the first time about this Indian mathematician and my searches led me to his article about so-called Copernicus magic squares. Finally, I've made an animation that contains some considerations on the theme of Copernicus magic square and I want to present it here (I strongly recommend you to watch this video in a full-screen mode):

http://www.youtube.com/watch?v=uMp6NwoEHVQ&feature=youtu.be
Post 17 Jul 2015, 00:04
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Tyler



Joined: 19 Nov 2009
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Location: NC, USA
That's cool. Nice video. How'd you make it? Also, any clue how he came up with that? Surely he didn't brute force it. My first guess would be some weird application of group theory, but he's way, way before group theory.
Post 17 Jul 2015, 05:00
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
Tyler wrote:
Also, any clue how he came up with that? Surely he didn't brute force it. My first guess would be some weird application of group theory, but he's way, way before group theory.
I don't think that he (D.R. Kaprekar) needed to use such advanced mathematical "apparatus" as Group Theory. Here rather the main role plays natural, inborn ability of people from the Indian Peninsula to perform all necessary calculations in mind. I've mentioned here a great role of intuition and imagination in their mathematical discoveries.

Also, there exist some methods of creation of magic squares explored and explained very well by French mathematicians of the 17th century - La Loubere, Bachet and La Hire.
Post 17 Jul 2015, 22:45
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
When I'm fed up with foggy autumn days I create sunny landscapes that serve me as a background for mathematical problems and their solutions. Wink

Image

Image
Post 03 Nov 2015, 13:47
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
Just for the sake of completeness (every question should have corresponding answer Wink ):

Image

Image

Use of inequality between weighted arithmetic mean and weighted geometric mean:

Image
Post 03 Nov 2015, 13:47
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
Image
Post 03 Nov 2015, 13:48
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
YONG wrote:
To keep the proof simple, I would start from

[sqrt(a) - sqrt(b)]^2 >= 0

where a & b are non-negative real numbers.

Then we have

a - 2 sqrt(ab) + b >= 0

a + b >= 2 sqrt(ab)

(a + b)/2 >= sqrt(ab)

So, A.M. >= G.M.


BTW, the background images of your solutions are really nice.

Wink
My proof is quite simple too. The only thing that makes it longer (visually) is that I show everything step-by-step and have added some descriptive explanations in English.

BTW, in the middle of the proof we get the inequality "(a + b)^2 >= 4ab" for every real a,b that is frequently used in solutions of other inequalities.

Smile
Post 03 Nov 2015, 13:49
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
A graph presenting a kind of "genetic relationship" between basic inequalities.

The symbols of a form "/<operator> f(a,b)" next to the arrows symbolize the operation that is to be done on the both sides of the given inequality to transform it into another inequality. For example, symbol "/ + 4ab" means that we have to add 4ab to both sides of inequality.

Image
Post 03 Nov 2015, 13:50
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
YONG wrote:
Yeah, your proof gives some spin-off inequalities which are very useful.

BTW, I think you should ask a moderator to extract all these math posts and keep them in a general math thread. You started one some time ago, right?

Wink
I have copied most of the posts to the current thread. Smile
Post 03 Nov 2015, 14:38
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
Image
Post 18 Nov 2015, 22:38
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tthsqe



Joined: 20 May 2009
Posts: 721
(x-1)/(x+1) is increasing for x>1
(a/b)^p > (a/b)^q > 1 by hypothesis
Post 19 Nov 2015, 00:48
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