ganitkoumudi

  रेखागणितम् - 6-10 मराठी भाषांतर - एआय सारांश  अथ षष्ठं क्षेत्रम् । If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another. तत्र यस्य त्रिबुजस्य कोणद्वयम समानं तत्कोंसंबन्दि भुजद्वयमपि समानं भवति । अत्रोपपत्ति । तत्र अबजत्रिभजे बजकोणौ समानौ । अबं अजमपि समानम् । यदि भुजद्वयं समानम न भवति एको भुजोऽधिक: स्यात्तदा अधिकभुज: अजं कल्पित: । बअसमानं जदं भिन्नं कृत्वा बदरेखा कार्या । अजबत्रिभुजे अबभुजो बजभुज; अबजकोण: दबजत्रिभुजस्य दजभुजेन जबभुजेन दजबकोणेन समान: । एवं बृहत्त्रिभुजं लघुत्रिभुजसमानं जातम् । तदिदमनुपपन्नम् । बृहत्क्षेत्रं लघुक्षेत्रेण कथं समानं भविष्यति । तस्मात् अजं अबं समानम् । तदेवमुपपन्नं कोनद्वयसाम्ये तत्सक्तभुजद्वयसाम्यं भवतीति ।। तत्र यस्य त्रिबुजस्य कोणद्वयम समानं तत्कोंसंबन्दि भुजद्वयमपि समानं भवति । अत्रोपपत्ति । तत्र अबजत्रिभजे बजकोणौ समानौ । अबं अजमपि समानम् । यदि भुजद्वयं समानम न भवति एको भुजोऽधिक: स्यात्तदा अधिकभुज: अजं कल्पित: । बअसमानं जदं भिन्नं कृत्वा बदरेखा कार्या । अजबत्रिभुजे अबभुजो बजभुज; अबजकोण:...

  भद्रगणितम् - Magic squares- Introduction - एआय विश्लेषण Introduction   Classification of Magic squares   Purpose of studying (as stated by Narayana)   Kacchaputa of N̄agarjuna (c.100 BCE)   Sarvatobhadra of Varahamihira (550 CE)   TheTuraga gati method of obtaining magic squares   Possible no.of 4×4 PD squares (with elements1,2...16)?    Ancient Indian method for odd squares   Kuttaka and magic squares   Properties of 4×4 magic squares   Construction of magic squares using these properties     Introduction Background and Relevance  Today there is a lot of discussion going on all around the world to see how to make mathematics learning more interesting.   As far as arithmetic is concerned, certainly one way to make it interesting is to introduce the topic of Magic Squares—called Bhadra-ganita in Indian Mathematics.   The nomenclatures terms from the fact it was considered to fetch bhadra—all round pros...

  Magic Squares in Indian Tradition- AI commented     In all magic squares, it is through arithmetic progression . . .   By those desirous . . . the first term and the common difference have to be determined.   As many as the number of boxes in the square will be equal to the number of terms (n2).                     Popularity of Magic squares in India   The first chapter of Srinivasa Ramanujan’s Notebooks is on Magic Squares. It is said to be "much earlier than the remainder of the notebooks”.    T.Vijayaraghavan,in his article on Jaina Magic Squares (1941)notes: “The author of this note learnt by heart at the age of nine the following pan-diagonal square which was taught to him by an elderly person who had not been to school at all.”  8 11 2 13 1 14 7 12 15 4 9 6 10 5 16 3         This clearly indicates the popularity of Magic...

  Narayan Pandit -Turagagati Method - AI comments     Sarvatobhadra of Varahamihira → Narayana’s This square Varahamihira’s +    2  3    5  8   5  8    2   3   4  1    7   6   7  6    4   1      8  0    8   0   0  8    0  8   0  8    0   8   8  0    8   0         Narayana’s   It is to be noted that all the three squares are pan-diagonal.   This belongs to a class of 4×4 pan-diagonal magic squares studied by Narayan. a Pandita in Ganitakaumudı (c.1356).   We’ll see later that there are 384 possible ways of constructing such (4 × 4) magic squares.    10  3    13   8   5  16    2...

  Narayan Pandit - 384 pan-diagonal magic squares  Possible no. of 4 × 4 PD squares (with elements 1,2. . . 16)?   1    8  13  12  14    11  2   7  4   5  16  9  15   10  3  6  ----    1  12 13   8   15  6 3   10   4  9 16   5   14  7 2   11           Narayana now poses the question:   Having displayed 24 pan-diagonal 4×4 magic squares,with the top left entry being 1, Narayana states:  Thus there are 384 possibilities in a magic square . . .   This has been proved by B.Rosser and R.J.Walker(1938);Much simpler proof was provided by T. Vijayaraghavan (1941).     Anci...

  Narayan Pandit- Kuttaka Magic Square - AI Comments    Kuttaka and magic squares   Given the magic sum S and the order of the magic square n,the first thing to be done to construct the magic square is to obtain the  defined by(a, d).   Having obtained (a, d ), the   having n2 elements is constructed and this will be used to fill in n × n square. Narayana makes use of the following kuttakara to obtain (a, d). 2 1 2  nS = n 2{a+(a+(n −1)d)} (1) or S = na+ n (n2 −1)d (2) 2   It is well known that in a kuttakara problem there exists an infinite number of integral solutions for (a, d) if S is divisible by the GCD of    n,   n   (n2 − 1) . In other words, S should be divisible by n when n is 2  odd, and by n for n even. 2 Suresh, this is a fascinating convergence of  Diophantine reasoning (Kuttaka)  and  magic square construction , as presented by Narāya...

  Narayan Pandit - Constructing 4x4 Magic Squares - AI Comments  Narayana’s example: Construct a 4×4 magic square with S=40.Now we have the equation 40 = 4a + 30d which is satisfied by the pairs (a, d ) = (−5, 2) (10, 0) (25. − 2), and so on.                         Properties of 4 × 4 pan-diagonal magic squares Property 1: Let M be a pan-diagonal 4 × 4 magic square with entries 1, 2, . . . , 16, which is mapped on to the torus by identifying opposite edges of the square. Then the entries of any 2x2 sub-square formed by consecutive rows and columns on the torus add up to 34.    1  12 13   8   15  6  3   10   4    9  16   5   14  7  2   11     Property 2: 1 + 12 + 15 + 6 = 1 + 12 + 14 + 7 = 34 1+15+8+10=4+14+9+7=...

  Natrayan Pandit- Magic Square Methos - AI Comments   Narayana's Example 1: 4x4 Magic Square with Sum 40 Narayana takes the sequence 1, 2, 3,4 as the base sequence  (mulapankti),  which is also called the first sequence; and the sequence 0, 1, 2, 3 as the other sequence  (parapankti),  which is also called the second sequence. The sum of the first sequence is 10. When this is subtracted from 40, or the desired magic sum  (phala),  we get 30. When this is divided by the sum of the second sequence, namely 6, we get 5 as the factor  (guna).  Multiplying each of the terms of the second sequence by this factor, we get the product sequence  (gunapankti)  0, 5, 10, 15. From these sequences, Narayana forms the covered  (chadya)  and the coverer  (chadaka)  squares as shown in Fig. 1. नारायण   पंडित मूलपङ्क्ति  =  १ , २ , ३ , ४  = १० ,  परापङ्क्ति  =  ० , १ , २ , ३  =  ६ ...

  Narayan Pandit-Folding Method for Magic Square - AI Comments   In 1938, Rosser and Walker proved that this is in fact the exact number of 4x4 pan-diagonal magic squares with entries 1, 2, ..., 16.  Vijayaraghavan (1941) gave a much simpler proof of this result, which have been outlined by Sridharan and Srinivas (2011, pp.389-391),  Narayana goes on to discuss general methods of construction of magic squares depending upon whether the square is  samagarbha  (doubly-even),  visamagarbha  (singly-even) or  visama  (odd). In the case of  samagarbha  and  visama  squares, apart from discussing the traditionally well-known methods of construction (indicated for instance in Pheru's work earlier),  Narayana presents an entirely new method known as  samputa-vidhi  (method of folding). This is a general method of construction of magic squares by composing or folding two magic squares constructed suitably Naraya...