Universal Bimagic Squares and the day 10th October 2010 (10.10.10)
aa r X i v : . [ m a t h . HO ] O c t Universal Bimagic Squares andthe day 10 th October 2010 (10.10.10)
Inder Jeet Taneja
Departamento de Matem´aticaUniversidade Federal de Santa Catarina88.040-900 Florian´opolis, SC, Brazil. ∼ taneja Abstract
In this short note we have produced for the first time in the history differentkinds of universal bimagic squares. This we have made using only the digits 0,1 and2. The universal bimagic squares of order × and × are with the digits 0and 1. The universal bimagic square of order × is with the digits 0, 1 and 2.It is interesting to note that the day 0ctober 10 have only the digits 0 and 1 if weconsider it as 10.10.10. If we consider the date as 10.10.2010, then this has thedigits 0, 1 and 2. In this work we shall present universal bimagic squares of order 8 × ×
16 havingonly the digits 0 and 1. A universal bimagic square of order 9 × Here below are some definitions. • Magic square
A magic square is a collection of numbers put as a square matrix, where the sum ofelements of each row, sum of elements of each column or sum of elements of each of twoprincipal diagonals are the same. For simplicity, let us write this sum as S1 . • Bimagic square
Bimagic square is a magic square where the sum of square of each element of rows, columnsor two principal diagonals are the same. For simplicity, let us write this sum as S2 . • Universal magic square
Universal magic square is a magic square with the following properties:1i)
Upside down , i.e., if we rotate it to 180 degrees, it remains magic square again;(ii)
Mirror looking , i.e., if we put it in front of mirror or see from the other side ofthe glass, or see on the other side of the paper, it always remains the magic square.
It is interesting to note that at 10 hours, 10 minutes and 10 seconds of the day 10, month10 and the year 10 have only the digits 1 and 0, i.e., 10-10-10-10-10-10. Let us divide itin two parts, i.e., 101010 – 101010. Thus we have two equal blocks of six algarisms. If wego only for the day, these digits repeats on others days too, such as01-10-10; 01-01-10; 10-01-10, 11-10-10, etc.If we go on hours, minutes and seconds we have many combinations of six algarismsonly with the digits 0 and 1. • × – Universal bimagic square of binary digits 0 and 1 We can make 2 × × × × × = 64 different numbers of six algarisms withthe digits 0 and 1. Also, we can write 64 = 8 ×
8. Here below is a universal bimagicsquare of order 8 × × × × – Universal bimagic square of binary digits 0 and 1 Instead, considering six algarisms using only the digits 0 and 1, if we consider eightalgarisms using we can make 2 = 256 different numbers only with the digits 0 and 1.Also we we can write 256 as 16 ×
16. Here below is a universal bimagic square of order16 ×
16 with these 256 different numbers made from the digits 0 and 1:S1:=88888888S2:=897867554657688Also we have sum of each block of 4 × × Instead, considering the year as 10, if we consider it as 2010, then we have three algarisms0, 1 and 2. These digits happens on other days too, such as02.01.2010; 02.02.2010; 20.10.2010; 02.10.2010; 12.10.2010; 2.10.2010, etc.Still there are many other dates having only the digits 0, 1 and 2.3 × – Universal bimagic square of digits 0, 1 and 2 We can make exactly 81 different numbers having four algarisms from the three digits 0,1 and 2, i.e, 3 × × × ×
9. Here below is a universalbimagic square of order 9 × × × References [1] .[2] http://recmath.org/Magic Squares . 43] I.J. TANEJA – DIGITAL ERA: Magic Squares and 8 th May 2010 (08.05.2010), http://arxiv.org/ftp/arxiv/papers/1005/1005.1384.pdf .[4] I.J. TANEJA – ERA DIGITAL E 50 ANOS DA UFSC,