Upside Down Magic, Bimagic, Palindromic Squares and Pythagoras Theorem on a Palindromic Day - 11.02.2011

arXiv:1102.2394v2 [math.HO] 14 Feb 2011 Upside down Magic, Bimagic, Palindromic Squares and Pythagoras Theorem on a Palindromic Day - 11.02.2011 Inder Jeet Taneja Departamento de Matem´atica Universidade Federal de Santa Catarina 88.040-900 Florian´opolis, SC, Brazil. e-mail: taneja@mtm.ufsc.br http://www.mtm.ufsc.br/∼taneja Abstract In this short note we have produced different kinds of upside down magic squares based on a palindromic day 11.02.2011. In this day appear only the algorisms 0, 1 and 2. Some of the magic squares are bimagic and some are palindromic. Magic sums of the magic squares of order 3×3, 4×4 and 5×5 satisfies the Pythagoras theorem. Three different kinds of bimagic squares of order 9×9 are also produced. The bimagic square of order 9×9 with 8 digits is palindromic numbers. We have given bimagic squares of order 16×16 and 25×25, where the magic sum S1 in both the cases is same. In order to make these magic squares upside down, i.e., 1800 degree rotation, we have used the numbers in the digital form. All these magic square are only with three digits, 0, 1 and 2 appearing in the day 11.02.2011. 1 Introduction It interesting to observe that the day 11.02.2011 is palindromic and has only three digits 0, 1 and 2. A similar kind of palindromic day shall also appear next year 21.02.2012 having the same three digits. In this paper our interest is to produce upside down magic squares, bimagic squares and palindromic magic squares using only these three algorisms, 0, 1 and 2. A similar kind of study can be seen in another author’s work on the day October 10, 2010 [5]. Using these three digits we have [8] made equivalence with classical magic squares, the one is ”Lo-Shu” magic square of order 3×3 and another is ”Khajurao” magic squares of order 4×4. Before we proceed, here below are some basic definitions: (i) A magic square is a collection of numbers put as a square matrix, where the sum of element of each row, sum of element of each column and sum of each element of two principal diagonals have the same sum. For simplicity, let us write it as S1. (ii) Bimagic square is a magic square where the sum of square of each element of rows, columns and two principal diagonals are the same. For simplicity, let us write it as S2. 1 (iii) Upside down, i.e., if we rotate it to 1800 degree it remains the same. (iv) Mirror looking, i.e., if we put it in front of mirror or see from the other side of the glass, or see on the other side of the paper, it always remains the magic square. (v) Universal magic squares, i.e., magic squares having the property of upside down and mirror looking are considered universal magic squares. In this short note we have produced different kinds of upside down magic squares using only the algorisms 0, 1 and 2. Some of the magic squares are bimagic and some are palindromic. Magic sum of the magic squares of order 3×3, 4×4 and 5×5 satisfies the Pythagoras theorem. Bimagic squares of order 9x9 are produced with 4, 6 and 8 digits. The bimagic square of order 9×9 with 8 digits is palindromic while with 6 digits is a combination of palindromic numbers. We have given bimagic squares of order 16×16 and 25×25, where the magic sum S1 is the same. In order to make these magic squares upside down we have used the numbers in the digital form. All these magic square are only with three digits, 0, 1 and 2. In order to do so, we have used the numbers in the digital form: These digits generally appear in watches, elevators, etc. We observe that the above three digits are rotatable to 1800, and remains the same. These also be considered as universal, because in the mirror 2 becomes 5, while 0 and 1 remains the same. In these situations the magic sums are different. This we leave to reader to verify. 2 Upside Down Magic Squares and the Pythagoras Theorem In this section we shall present magic squares of order 3×3, 4×4 and 5×5 having only the three digits 0, 1 and 2 in the digital form. Interesting the magic sum S1, in this case satisfies the Pythagoras theorem. • Magic squares of order 3×3 Here below are two magic squares of order 3×3 with S13×3 := 33 and S13×3 := 3333 respectively. The first one is with two digits combinations while the second one is with four digits combinations: 2 We observe from the second magic square that it is palindromic. In order to have upside down we have considered 110, 220 as 0110, 0220 to be symmetry in the result. • Magic squares of order 4×4 Here below is a magic square of order 4x4 with S14×4 := 4444 • Magic squares of order 5×5 Here below is a magic square of order 5×5 with S15×5 := 5555 The above magic square is pan diagonal. 3 2.1 Pythagoras Theorem From the above magic squares of orders 3×3, 4×4 and 5×5 with four digits, we have the following result: (S13×3)2 + (S14×4)2 = (S15×5)2 , i.e., 33332 + 44442 = 55552 i.e., 11108889 + 19749136 = 30858025. This means that if we consider square of any line, or column or principle diagonal, from one of the above magic squares we sh

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