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s at TathaGat Delhi! Display replies flat, with oldest first Pythagorean Soliloquy by Total Gadha - W e dne sday, 31 De ce m be r 2008, 02:07 PM
Years ago, my Italian language teacher and my classmates learning Italian used to question me how I could pick up the language so fast. “Perche sono un insegnante di matematica,” was my standard answer. That would leave them baffled. What was the relation between language and mathematics? Little do my classmates or many other students, especially our C AT aspirants, understand that language and mathematics are the two sides of the same coin; both consist of symbols connected through logic. And more often than not, even the logic they follow is the same. The symbols for both were gradually developed. The mathematician Euler was responsible for our common, modern-day use of many famous mathematical notations—for example, f(x) for a function, e for the base of natural logs, i for the square root of –1, Π for pi, Σ for summation. Shakespeare is credited with coining nearly 1700 words in the English language. These are our symbols. The logic was also developed gradually; the grammar, mathematical proofs, critical reasoning, logical reasoning, etc. form the parts of logic that bind those symbols. Therefore, if you are a logical creature, chances are that you would not face difficulty understanding either of these subjects, provided you are comfortable with the symbols- English vocabulary or mathematical notations, whatever the case maybe. The progression of thoughts of a writer is not markedly different from that of a mathematician. While reading a good age one can automatically guess the next paragraph. In the same manner, a mathematician proceeds from one thought to another in a logical manner, removing obstacles in his path to discovery and finding answers to the problems one after another.
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Re: Pythagorean Soliloquy by Gowri Nandana - W e dne sday, 31 De ce m be r 2008, 02:22 PM Awe som e article ........ Show parent | Reply
Re: Pythagorean Soliloquy by Gowri Nandana - W e dne sday, 31 De ce m be r 2008, 02:35 PM Happy Ne w ye ar to the supe r TG te am . May God gift u with 48 hours a day to m anage ur wonde rful site
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Re: Pythagorean Soliloquy by Dagny Taggart - Thursday, 1 January 2009, 05:28 PM A ve ry Hppy Ne w Ye ar to you too, Gowri. Show parent | Reply
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Re: Pythagorean Soliloquy by sande e p som avarapu - Thursday, 1 January 2009, 08:55 PM Sir ji...W hat an e x plantion sir ji..... ------------one conce pt can change your thought proce ss I wish you and TG fam ily a ve ry happy Ne w ye ar. Show parent | Reply
Re: Pythagorean Soliloquy by rajat shuk la - Friday, 2 January 2009, 08:28 PM happy ne w ye ar...to all of u.. Show parent | Reply
Re: Pythagorean Soliloquy by rashi agarwal - Friday, 2 January 2009, 11:59 PM he llo TG sir, A ve ry happy Ne w Ye ar to you...Gre at article ...... sir, i hav a little confusion.To calculate the no. of right triangle s with a give n le ngth N as one of its le g, N= 2 a x p b x q c x ................. The form ula is [(2a -1)(2b+1)(2c+1)....-1 ]/ 2 But sir if we want to calculate with 25 as the le ngth of the le g. the n a =0. the n whole te rm will be -ve . this m e thod will not be use d for the odd num be rs..ple ase sir corre ct m e if I am wrong. rashi Show parent | Reply
Re: Pythagorean Soliloquy by Total Gadha - Saturday, 3 January 2009, 02:18 AM Hi R ashi, Don't conside r the te rm 2a - 1 if N is an odd num be r, i.e . no powe r of 2 is pre se nt. Total Gadha Show parent | Reply
Re: Pythagorean Soliloquy by Dipanjan Biswas - Sunday, 4 January 2009, 11:48 PM Thank you sir a lot for such a nice article ..It re ally foste rs ne ural ne twork of thoughts..Sim ultane ously it cove rs up num be r syste m ...Thank s once again.. Happy Ne w Ye ar to all of TG fam ily TG rock zzz Dipanjan Show parent | Reply
Re: Pythagorean Soliloquy by Tuhin Bane rje e - Monday, 5 January 2009, 06:24 PM Hi TG, R e ally a awe som e article . Thank s a zillion m am . Show parent | Reply
Re: Pythagorean Soliloquy by gadha abc - Monday, 5 January 2009, 06:49 PM Each se nte nce spe ak s the inte llige nce .. Happy ne w TG sir and Mam . Sir, can u ple ase little e laborate how that form ula u de rive N+5C 5 for the dice proble m . R e gards Show parent | Reply
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Re: Pythagorean Soliloquy by him anshu m ishra - Sunday, 11 January 2009, 10:49 PM hi,tg This article is supe rb...........all the com m e nts are use le ss for this nugge t of wisdom . Show parent | Reply
Re: Pythagorean Soliloquy by Sharadha Kuntum alla - Monday, 26 January 2009, 12:57 AM He llo TG Sir, I want to copy the article in a word docum e nt so that I can print it, highlight im portant points and k e e p re vising it. But the conte nt is not ge tting aligne d prope rly on the word docum e nt. I won't find this difficulty whe n I am copying ve rbal article s/e x e rcise s. C an you ple ase he lp m e with a printable ve rsion of the se wonde rful article s. Shadh Show parent | Reply
Re: Pythagorean Soliloquy by Total Gadha - Monday, 26 January 2009, 11:09 AM Hi Sharadha, Many of the se article s are im age s. R ight click on the m , 'copy' the m and paste the m in paint and the n save the m as im age s. The n you can print the m out. Total Gadha Show parent | Reply
Re: Pythagorean Soliloquy by sum it jam wal - Thursday, 29 January 2009, 12:50 AM hi sir, as you state d in the article if i tak e 25 As hypote nuse the n the re are 25=5^2 so ((2*2+1)-1)/2 = 2 bt i cud find only 5^2=3^2 +4^2 confuse d Show parent | Reply
Re: Pythagorean Soliloquy by Total Gadha - Thursday, 29 January 2009, 10:25 AM Hi Sum it, You are tak ing 5 or 25 as hypote nuse ? If you are tak ing 25, the n you ne e d to tak e 625 = 25 2. Total Gadha Show parent | Reply
Re: Pythagorean Soliloquy by Anurag R ai - Sunday, 1 Fe bruary 2009, 02:06 PM Nice and Brilliant Show parent | Reply
Pythagorean Soliloquy by Abhishe k Bansal - Monday, 2 Fe bruary 2009, 02:17 AM As i was atte nding a class on Ge om e try. I was told that the re the re is one triple t such as (20,21,29). I starte d se arching and wante d to find a way to ge t to the num be r of triple ts if i am give n one of the side s. A side can be the longe st side (i.e . hypote nuse ) or base /pe rpe ndicular.
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MBA| CAT| CAT 2011| CAT 2012| CAT Online| MBA 2012| MBA Entrance Exams| CAT Test| Preparation| CAT Questions i will say its by she e r coincide nce , that som e one told m e about this site today whe n i was tak ing the class - and in that class i took the case of one of the side s be ing 5^11. i k ne w a way to find whe n 5^11 is not the longe st side or any num be r for that m atte r whe n it is not the longe st side .. if we are look ing for inte gral solutions the n (a-b)*(a+b) should be e ve n or odd... so 11 triangle s are possible . I wrote a code in Matlab (just to e nsure that i am not the one doing the calculation) - i took 5^(n) ... n starting from 1,2,3,..... 12,13... at n = 15 it starte d showing som e e rrors re late d to m e m ory. I re ally ne e d to che ck the code , and the re sults.. be cause at n =12 it give s m e 16 triangle s which are possible whe n 5^12 is the hypote nuse and 5^13 give s m e 60 triangle s. what i saw till 5^11.. the answe r will be the powe r itse lf. and 3^n and 7^n won't give m e any triangle s. and 31^n and 37^n.. will start giving m e triangle s for som e value of n > 2,3 so can u che ck the validity. I can be wrong. i just have m y code and the re sults and those i will post tom orrow.
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Re: Pythagorean Soliloquy by Abhishe k Bansal - Monday, 2 Fe bruary 2009, 02:42 AM It is quite possible that the num be r sudde nly incre ase s be cause of the pre cision e rrors. Show parent | Reply
Re: Pythagorean Soliloquy by Harish Bansal - W e dne sday, 4 Fe bruary 2009, 12:22 AM Hi TG, I trie d to de rive the sam e form ula that u have writte n at the top by using the sam e e x am ple of 60^2. I got a diffe re nt re sult. My re sult is: {(a-1)(b+1)(c+1)(d+1)....-1}/2 Ple ase clarify Show parent | Reply
Re: Pythagorean Soliloquy by sum it jam wal - W e dne sday, 4 Fe bruary 2009, 08:44 AM
Thank s TG
..m y m istak e Show parent | Reply
Re: Pythagorean Soliloquy by Total Gadha - W e dne sday, 4 Fe bruary 2009, 09:25 AM Hi Harish, Maybe if you can e x plain how you de rive d the form ula, I can point out the m istak e . Total Gadha Show parent | Reply
Re: Pythagorean Soliloquy by alsadra @TG - Monday, 16 Fe bruary 2009, 04:01 PM sir i 've be e n re ading this article for m ore than 90 m inute s. had to im agine a lot. can u ple ase e x plain m e how u calculate d the no of ways of e x pre ssing 65 as a sum of two square s towards the e nd of the article . the article is ve ry inform ative and ve ry de nse in inform ation. thank s sir... Show parent | Reply
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MBA| CAT| CAT 2011| CAT 2012| CAT Online| MBA 2012| MBA Entrance Exams| CAT Test| Preparation| CAT Questions Re: Pythagorean Soliloquy by srihari sank araram an - Thursday, 19 Fe bruary 2009, 05:49 AM He llo Sir, She e r brilliance ... I was not able to prove the last the ore m . C ould you te ll m e how you cam e to this conclusion.(n+5)c5? O nce again,Kudos to TG. Show parent | Reply
Re: Pythagorean Soliloquy by sye d haque - W e dne sday, 25 Fe bruary 2009, 05:53 PM Hi TG, Can you please explain why you have divided the number of factors by 2? As two different pairs can give different values of a and b and will result in two right triangles.Sir please explain me this.
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Re: Pythagorean Soliloquy by am rit jajodia - Sunday, 1 March 2009, 12:28 AM Hi, Just one word for the article ..... Awe som e !!!!! I am not cle ar about the no of distinct outcom e form ula. Just for the case if N = 2 the n no of instinct outcom e possible is nos from 2 to 12. that is 11 outcom e s but with the form ula i ge t 21. C an you ple ase clarify on this. And m ore ove r can you ple ase put up the answe r for those thre e que stions also. Show parent | Reply
Re: Pythagorean Soliloquy by Abhishe k Bansal - Sunday, 1 March 2009, 10:46 AM you are talk ing about the sum ... whe n you se e 2 to 12.... se e the total num be r of outcom e s whe n you throw two dice will be 36 if the dice are not ide ntical to e ach othe r.... 1,2 is diffe re nt from 2,1 but whe n you throw two ide ntical dice toge the r.... 1,2 is sam e as 2,1 the re is no diffe re nce as you won't be able to distinguish... so those k ind of case s will be 21... 1,1 2,2 3,3 4,4 5,5 6,6 if you tak e this case s out of the sam ple space .. the re are 30 m ore case s le ft... now the se 30 case s have (p,q) and (q,p) whe re p is not e qual to q. so the se case s are counte d twice .... 30/2 + 6 = 15 +6 = 21case s Show parent | Reply
Re: Pythagorean Soliloquy by yoge sh bansal - Tue sday, 23 June 2009, 02:47 PM hi...answe r of first 2 que stions out of last 3 que stions... 1.) 48 2.) 2 7
Am i correct??? Tg plz let me knw.. Show parent | Reply
Re: Pythagorean Soliloquy by Mohit Bham bri - Friday, 4 Se pte m be r 2009, 10:06 PM
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Re: Pythagorean Soliloquy by Mohit Bham bri - Friday, 4 Se pte m be r 2009, 10:52 PM
2) I too think it is 2^7. For a num be r to be hypote nuse it m ust have a prim e no of form 4n+1 in its factors. So if we start with a num be r of form 5x (..) so we have to ge t re m aining 5 triangle s from side as 5x (..) which m e ans [(2a-1)(2b+1)(2(1)+1)-1 ]/2 =5 cause we alre ady le t 5 be a factor thats y the 2(1)+1 *if the re is a powe r of2 which give s 3x som e thing =11 not possible he nce having a single 4n+1 factor is rule d out sim ilarly having 2 4n+1 factors is rule d out i.e 13, 5 at the sam e tim e the re fore only possibility to have 6 triangle s is with it as SIDE O NLY not hypote nuse . by logic of 1) we ge t N= 2^7 C orre ct m e if im wrong Show parent | Reply
Re: Pythagorean Soliloquy by Mohit Bham bri - Friday, 4 Se pte m be r 2009, 10:54 PM
The re is som e proble m in fram ing que stion 3 as it doe snt spe cify 3 traingle s having inte gral side s of e qual are as. Show parent | Reply
Re: Pythagorean Soliloquy by srinivasan ravi - Friday, 11 Se pte m be r 2009, 12:20 AM hi m ohit, can u e x plain the 2nd que stion m ore cle arly..thank s.. Show parent | Reply
Re: Pythagorean Soliloquy by R aju Singh - Monday, 28 Se pte m be r 2009, 02:34 AM @ Mohit C ould we hav som e sim ple approach for sol of 2 Q ue stion ask e d: Sm alle st num be r for 6 distinct right triangle for inte ge r side s: [{(2a-1)(2b+1)(2c+1)}...-1]/2=6 (2a-1)(2b+1)(2c+1)=2x 6+1 (2a-1)(2b+1)(2c+1)=13 As R HS is 13 which is prim e num be r so any of LHS te rm m ust have 13 as one of the num be r. For num be r to be m inim um only (2a-1) m ust be e qual to 13 and re st all te rm s as 1 The re fore , (2a-1)=13, (2b+1) =1 and (2c+1)=1 e tc.. he nce a=7 and b=c=...=0 And the re by 2^7 as the sm alle st num be r for rt triangle s with inte ge r side le ngths. Mohit what u hv e laborate d I'm not ge tting plz clarify it. Thank s Show parent | Reply
Re: Pythagorean Soliloquy by De e pak Kum ar - Saturday, 9 O ctobe r 2010, 12:31 PM Hi TG Sir, R e ally a gre at artical. I lik e d the way you have writte n the artical, logically com ing to the ne x t ste p, the way you told in the pre face of the artical.
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Re: Pythagorean Soliloquy by abhishe k abhi - Saturday, 1 O ctobe r 2011, 08:46 PM sir in the first proble m ,,to find out odd pairs 3*3 is done ,,sir could you ple ase e x plain m e how to find odd pairs in those factor pairs Show parent | Reply
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Re: Pythagorean Soliloquy by Kam al Joshi - Friday, 7 O ctobe r 2011, 05:40 AM Sir, Ple ase e x plain, how did you de rive (2^2+1^2)(2^2+3^2)(2^2+1^2)(2^2+3^2) = 63^2+16^2 = 33^2+56^2??? Show parent | Reply
Re: Pythagorean Soliloquy by Kam al Joshi - Friday, 7 O ctobe r 2011, 04:22 PM O k got it, sir!!! You de rive d diffe re nt com binations of 65^2 by (a^2+b^2)(c^2+d^2)= (ac+bd)^2+(ad-bc)^2 = (ac-bd)^2+(ad+bc)^2. C orre ct? Show parent | Reply
Re: Pythagorean Soliloquy by prabhat tane ja - W e dne sday, 17 O ctobe r 2012, 05:38 PM one word.. SUPER B.. I lik e d the article ove rall, but the footnote m ade m y day.. Thank s to Mr Mahajan for the unique solution and TG for such fabulous a post. Show parent | Reply
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s at TathaGat Delhi! Display replies flat, with oldest first Pythagorean Soliloquy by Total Gadha - W e dne sday, 31 De ce m be r 2008, 02:07 PM
Years ago, my Italian language teacher and my classmates learning Italian used to question me how I could pick up the language so fast. “Perche sono un insegnante di matematica,” was my standard answer. That would leave them baffled. What was the relation between language and mathematics? Little do my classmates or many other students, especially our C AT aspirants, understand that language and mathematics are the two sides of the same coin; both consist of symbols connected through logic. And more often than not, even the logic they follow is the same. The symbols for both were gradually developed. The mathematician Euler was responsible for our common, modern-day use of many famous mathematical notations—for example, f(x) for a function, e for the base of natural logs, i for the square root of –1, Π for pi, Σ for summation. Shakespeare is credited with coining nearly 1700 words in the English language. These are our symbols. The logic was also developed gradually; the grammar, mathematical proofs, critical reasoning, logical reasoning, etc. form the parts of logic that bind those symbols. Therefore, if you are a logical creature, chances are that you would not face difficulty understanding either of these subjects, provided you are comfortable with the symbols- English vocabulary or mathematical notations, whatever the case maybe. The progression of thoughts of a writer is not markedly different from that of a mathematician. While reading a good age one can automatically guess the next paragraph. In the same manner, a mathematician proceeds from one thought to another in a logical manner, removing obstacles in his path to discovery and finding answers to the problems one after another.
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Re: Pythagorean Soliloquy by Gowri Nandana - W e dne sday, 31 De ce m be r 2008, 02:22 PM Awe som e article ........ Show parent | Reply
Re: Pythagorean Soliloquy by Gowri Nandana - W e dne sday, 31 De ce m be r 2008, 02:35 PM Happy Ne w ye ar to the supe r TG te am . May God gift u with 48 hours a day to m anage ur wonde rful site
Show parent | Reply
Re: Pythagorean Soliloquy by Dagny Taggart - Thursday, 1 January 2009, 05:28 PM A ve ry Hppy Ne w Ye ar to you too, Gowri. Show parent | Reply
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MBA| CAT| CAT 2011| CAT 2012| CAT Online| MBA 2012| MBA Entrance Exams| CAT Test| Preparation| CAT Questions Re: Pythagorean Soliloquy by Gul Gul - Thursday, 1 January 2009, 07:19 PM O ne of the be st ....! Show parent | Reply
Re: Pythagorean Soliloquy by sande e p som avarapu - Thursday, 1 January 2009, 08:55 PM Sir ji...W hat an e x plantion sir ji..... ------------one conce pt can change your thought proce ss I wish you and TG fam ily a ve ry happy Ne w ye ar. Show parent | Reply
Re: Pythagorean Soliloquy by rajat shuk la - Friday, 2 January 2009, 08:28 PM happy ne w ye ar...to all of u.. Show parent | Reply
Re: Pythagorean Soliloquy by rashi agarwal - Friday, 2 January 2009, 11:59 PM he llo TG sir, A ve ry happy Ne w Ye ar to you...Gre at article ...... sir, i hav a little confusion.To calculate the no. of right triangle s with a give n le ngth N as one of its le g, N= 2 a x p b x q c x ................. The form ula is [(2a -1)(2b+1)(2c+1)....-1 ]/ 2 But sir if we want to calculate with 25 as the le ngth of the le g. the n a =0. the n whole te rm will be -ve . this m e thod will not be use d for the odd num be rs..ple ase sir corre ct m e if I am wrong. rashi Show parent | Reply
Re: Pythagorean Soliloquy by Total Gadha - Saturday, 3 January 2009, 02:18 AM Hi R ashi, Don't conside r the te rm 2a - 1 if N is an odd num be r, i.e . no powe r of 2 is pre se nt. Total Gadha Show parent | Reply
Re: Pythagorean Soliloquy by Dipanjan Biswas - Sunday, 4 January 2009, 11:48 PM Thank you sir a lot for such a nice article ..It re ally foste rs ne ural ne twork of thoughts..Sim ultane ously it cove rs up num be r syste m ...Thank s once again.. Happy Ne w Ye ar to all of TG fam ily TG rock zzz Dipanjan Show parent | Reply
Re: Pythagorean Soliloquy by Tuhin Bane rje e - Monday, 5 January 2009, 06:24 PM Hi TG, R e ally a awe som e article . Thank s a zillion m am . Show parent | Reply
Re: Pythagorean Soliloquy by gadha abc - Monday, 5 January 2009, 06:49 PM Each se nte nce spe ak s the inte llige nce .. Happy ne w TG sir and Mam . Sir, can u ple ase little e laborate how that form ula u de rive N+5C 5 for the dice proble m . R e gards Show parent | Reply
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Re: Pythagorean Soliloquy by him anshu m ishra - Sunday, 11 January 2009, 10:49 PM hi,tg This article is supe rb...........all the com m e nts are use le ss for this nugge t of wisdom . Show parent | Reply
Re: Pythagorean Soliloquy by Sharadha Kuntum alla - Monday, 26 January 2009, 12:57 AM He llo TG Sir, I want to copy the article in a word docum e nt so that I can print it, highlight im portant points and k e e p re vising it. But the conte nt is not ge tting aligne d prope rly on the word docum e nt. I won't find this difficulty whe n I am copying ve rbal article s/e x e rcise s. C an you ple ase he lp m e with a printable ve rsion of the se wonde rful article s. Shadh Show parent | Reply
Re: Pythagorean Soliloquy by Total Gadha - Monday, 26 January 2009, 11:09 AM Hi Sharadha, Many of the se article s are im age s. R ight click on the m , 'copy' the m and paste the m in paint and the n save the m as im age s. The n you can print the m out. Total Gadha Show parent | Reply
Re: Pythagorean Soliloquy by sum it jam wal - Thursday, 29 January 2009, 12:50 AM hi sir, as you state d in the article if i tak e 25 As hypote nuse the n the re are 25=5^2 so ((2*2+1)-1)/2 = 2 bt i cud find only 5^2=3^2 +4^2 confuse d Show parent | Reply
Re: Pythagorean Soliloquy by Total Gadha - Thursday, 29 January 2009, 10:25 AM Hi Sum it, You are tak ing 5 or 25 as hypote nuse ? If you are tak ing 25, the n you ne e d to tak e 625 = 25 2. Total Gadha Show parent | Reply
Re: Pythagorean Soliloquy by Anurag R ai - Sunday, 1 Fe bruary 2009, 02:06 PM Nice and Brilliant Show parent | Reply
Pythagorean Soliloquy by Abhishe k Bansal - Monday, 2 Fe bruary 2009, 02:17 AM As i was atte nding a class on Ge om e try. I was told that the re the re is one triple t such as (20,21,29). I starte d se arching and wante d to find a way to ge t to the num be r of triple ts if i am give n one of the side s. A side can be the longe st side (i.e . hypote nuse ) or base /pe rpe ndicular.
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MBA| CAT| CAT 2011| CAT 2012| CAT Online| MBA 2012| MBA Entrance Exams| CAT Test| Preparation| CAT Questions i will say its by she e r coincide nce , that som e one told m e about this site today whe n i was tak ing the class - and in that class i took the case of one of the side s be ing 5^11. i k ne w a way to find whe n 5^11 is not the longe st side or any num be r for that m atte r whe n it is not the longe st side .. if we are look ing for inte gral solutions the n (a-b)*(a+b) should be e ve n or odd... so 11 triangle s are possible . I wrote a code in Matlab (just to e nsure that i am not the one doing the calculation) - i took 5^(n) ... n starting from 1,2,3,..... 12,13... at n = 15 it starte d showing som e e rrors re late d to m e m ory. I re ally ne e d to che ck the code , and the re sults.. be cause at n =12 it give s m e 16 triangle s which are possible whe n 5^12 is the hypote nuse and 5^13 give s m e 60 triangle s. what i saw till 5^11.. the answe r will be the powe r itse lf. and 3^n and 7^n won't give m e any triangle s. and 31^n and 37^n.. will start giving m e triangle s for som e value of n > 2,3 so can u che ck the validity. I can be wrong. i just have m y code and the re sults and those i will post tom orrow.
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Re: Pythagorean Soliloquy by Abhishe k Bansal - Monday, 2 Fe bruary 2009, 02:42 AM It is quite possible that the num be r sudde nly incre ase s be cause of the pre cision e rrors. Show parent | Reply
Re: Pythagorean Soliloquy by Harish Bansal - W e dne sday, 4 Fe bruary 2009, 12:22 AM Hi TG, I trie d to de rive the sam e form ula that u have writte n at the top by using the sam e e x am ple of 60^2. I got a diffe re nt re sult. My re sult is: {(a-1)(b+1)(c+1)(d+1)....-1}/2 Ple ase clarify Show parent | Reply
Re: Pythagorean Soliloquy by sum it jam wal - W e dne sday, 4 Fe bruary 2009, 08:44 AM
Thank s TG
..m y m istak e Show parent | Reply
Re: Pythagorean Soliloquy by Total Gadha - W e dne sday, 4 Fe bruary 2009, 09:25 AM Hi Harish, Maybe if you can e x plain how you de rive d the form ula, I can point out the m istak e . Total Gadha Show parent | Reply
Re: Pythagorean Soliloquy by alsadra @TG - Monday, 16 Fe bruary 2009, 04:01 PM sir i 've be e n re ading this article for m ore than 90 m inute s. had to im agine a lot. can u ple ase e x plain m e how u calculate d the no of ways of e x pre ssing 65 as a sum of two square s towards the e nd of the article . the article is ve ry inform ative and ve ry de nse in inform ation. thank s sir... Show parent | Reply
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MBA| CAT| CAT 2011| CAT 2012| CAT Online| MBA 2012| MBA Entrance Exams| CAT Test| Preparation| CAT Questions Re: Pythagorean Soliloquy by srihari sank araram an - Thursday, 19 Fe bruary 2009, 05:49 AM He llo Sir, She e r brilliance ... I was not able to prove the last the ore m . C ould you te ll m e how you cam e to this conclusion.(n+5)c5? O nce again,Kudos to TG. Show parent | Reply
Re: Pythagorean Soliloquy by sye d haque - W e dne sday, 25 Fe bruary 2009, 05:53 PM Hi TG, Can you please explain why you have divided the number of factors by 2? As two different pairs can give different values of a and b and will result in two right triangles.Sir please explain me this.
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Re: Pythagorean Soliloquy by am rit jajodia - Sunday, 1 March 2009, 12:28 AM Hi, Just one word for the article ..... Awe som e !!!!! I am not cle ar about the no of distinct outcom e form ula. Just for the case if N = 2 the n no of instinct outcom e possible is nos from 2 to 12. that is 11 outcom e s but with the form ula i ge t 21. C an you ple ase clarify on this. And m ore ove r can you ple ase put up the answe r for those thre e que stions also. Show parent | Reply
Re: Pythagorean Soliloquy by Abhishe k Bansal - Sunday, 1 March 2009, 10:46 AM you are talk ing about the sum ... whe n you se e 2 to 12.... se e the total num be r of outcom e s whe n you throw two dice will be 36 if the dice are not ide ntical to e ach othe r.... 1,2 is diffe re nt from 2,1 but whe n you throw two ide ntical dice toge the r.... 1,2 is sam e as 2,1 the re is no diffe re nce as you won't be able to distinguish... so those k ind of case s will be 21... 1,1 2,2 3,3 4,4 5,5 6,6 if you tak e this case s out of the sam ple space .. the re are 30 m ore case s le ft... now the se 30 case s have (p,q) and (q,p) whe re p is not e qual to q. so the se case s are counte d twice .... 30/2 + 6 = 15 +6 = 21case s Show parent | Reply
Re: Pythagorean Soliloquy by yoge sh bansal - Tue sday, 23 June 2009, 02:47 PM hi...answe r of first 2 que stions out of last 3 que stions... 1.) 48 2.) 2 7
Am i correct??? Tg plz let me knw.. Show parent | Reply
Re: Pythagorean Soliloquy by Mohit Bham bri - Friday, 4 Se pte m be r 2009, 10:06 PM
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MBA| CAT| CAT 2011| CAT 2012| CAT Online| MBA 2012| MBA Entrance Exams| CAT Test| Preparation| CAT Questions 1) 48 since we have to ge t 10 distinct [(2a-1)(2b+1)...-1]/2 =10 which give s (2a-1)(2b+1)...= 21 le t it be 3 and 7 tak ing 7 for powe rs of 2 and 3 for powe rs of 3 we ge t 2^4*3 =48 ..pls confirm
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Re: Pythagorean Soliloquy by Mohit Bham bri - Friday, 4 Se pte m be r 2009, 10:52 PM
2) I too think it is 2^7. For a num be r to be hypote nuse it m ust have a prim e no of form 4n+1 in its factors. So if we start with a num be r of form 5x (..) so we have to ge t re m aining 5 triangle s from side as 5x (..) which m e ans [(2a-1)(2b+1)(2(1)+1)-1 ]/2 =5 cause we alre ady le t 5 be a factor thats y the 2(1)+1 *if the re is a powe r of2 which give s 3x som e thing =11 not possible he nce having a single 4n+1 factor is rule d out sim ilarly having 2 4n+1 factors is rule d out i.e 13, 5 at the sam e tim e the re fore only possibility to have 6 triangle s is with it as SIDE O NLY not hypote nuse . by logic of 1) we ge t N= 2^7 C orre ct m e if im wrong Show parent | Reply
Re: Pythagorean Soliloquy by Mohit Bham bri - Friday, 4 Se pte m be r 2009, 10:54 PM
The re is som e proble m in fram ing que stion 3 as it doe snt spe cify 3 traingle s having inte gral side s of e qual are as. Show parent | Reply
Re: Pythagorean Soliloquy by srinivasan ravi - Friday, 11 Se pte m be r 2009, 12:20 AM hi m ohit, can u e x plain the 2nd que stion m ore cle arly..thank s.. Show parent | Reply
Re: Pythagorean Soliloquy by R aju Singh - Monday, 28 Se pte m be r 2009, 02:34 AM @ Mohit C ould we hav som e sim ple approach for sol of 2 Q ue stion ask e d: Sm alle st num be r for 6 distinct right triangle for inte ge r side s: [{(2a-1)(2b+1)(2c+1)}...-1]/2=6 (2a-1)(2b+1)(2c+1)=2x 6+1 (2a-1)(2b+1)(2c+1)=13 As R HS is 13 which is prim e num be r so any of LHS te rm m ust have 13 as one of the num be r. For num be r to be m inim um only (2a-1) m ust be e qual to 13 and re st all te rm s as 1 The re fore , (2a-1)=13, (2b+1) =1 and (2c+1)=1 e tc.. he nce a=7 and b=c=...=0 And the re by 2^7 as the sm alle st num be r for rt triangle s with inte ge r side le ngths. Mohit what u hv e laborate d I'm not ge tting plz clarify it. Thank s Show parent | Reply
Re: Pythagorean Soliloquy by De e pak Kum ar - Saturday, 9 O ctobe r 2010, 12:31 PM Hi TG Sir, R e ally a gre at artical. I lik e d the way you have writte n the artical, logically com ing to the ne x t ste p, the way you told in the pre face of the artical.
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Re: Pythagorean Soliloquy by abhishe k abhi - Saturday, 1 O ctobe r 2011, 08:46 PM sir in the first proble m ,,to find out odd pairs 3*3 is done ,,sir could you ple ase e x plain m e how to find odd pairs in those factor pairs Show parent | Reply
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MBA| CAT| CAT 2011| CAT 2012| CAT Online| MBA 2012| MBA Entrance Exams| CAT Test| Preparation| CAT Questions Re: Pythagorean Soliloquy by Just Gadha - W e dne sday, 5 O ctobe r 2011, 10:58 AM How do u ge t the total no. of solutions of the e quation a1+a2+a3+a4+a5=N as N+5 C 5 Show parent | Reply
Re: Pythagorean Soliloquy by Kam al Joshi - Friday, 7 O ctobe r 2011, 05:40 AM Sir, Ple ase e x plain, how did you de rive (2^2+1^2)(2^2+3^2)(2^2+1^2)(2^2+3^2) = 63^2+16^2 = 33^2+56^2??? Show parent | Reply
Re: Pythagorean Soliloquy by Kam al Joshi - Friday, 7 O ctobe r 2011, 04:22 PM O k got it, sir!!! You de rive d diffe re nt com binations of 65^2 by (a^2+b^2)(c^2+d^2)= (ac+bd)^2+(ad-bc)^2 = (ac-bd)^2+(ad+bc)^2. C orre ct? Show parent | Reply
Re: Pythagorean Soliloquy by prabhat tane ja - W e dne sday, 17 O ctobe r 2012, 05:38 PM one word.. SUPER B.. I lik e d the article ove rall, but the footnote m ade m y day.. Thank s to Mr Mahajan for the unique solution and TG for such fabulous a post. Show parent | Reply
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