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Author
MHajduk

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
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
02 Jul 2015, 16:54
Tyler

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

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
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
08 Jul 2015, 20:47
shoorick

Joined: 25 Feb 2005
Posts: 1605
Location: Ukraine
shoorick
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

Mr. Szurik
10 Jul 2015, 11:17
MHajduk

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
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".

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.
10 Jul 2015, 11:56
shoorick

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

Joined: 19 Nov 2009
Posts: 1216
Location: NC, USA
Tyler
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".
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.
12 Jul 2015, 06:33
MHajduk

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
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".
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.
12 Jul 2015, 09:42
Tyler

Joined: 19 Nov 2009
Posts: 1216
Location: NC, USA
Tyler
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.
14 Jul 2015, 23:21
MHajduk

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
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):

17 Jul 2015, 00:04
Tyler

Joined: 19 Nov 2009
Posts: 1216
Location: NC, USA
Tyler
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.
17 Jul 2015, 05:00
MHajduk

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
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.
17 Jul 2015, 22:45
MHajduk

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
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.

03 Nov 2015, 13:47
MHajduk

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
Just for the sake of completeness (every question should have corresponding answer ):

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

03 Nov 2015, 13:47
MHajduk

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
03 Nov 2015, 13:48
MHajduk

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
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.

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.

03 Nov 2015, 13:49
MHajduk

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
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.

03 Nov 2015, 13:50
MHajduk

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
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?

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

Joined: 30 Mar 2006
Posts: 6032
Location: Poland
MHajduk
18 Nov 2015, 22:38
tthsqe

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