Varino Magic Square

Varino Magic Square

Armahedi Mahzar (c) 2016

In my last blog I presented a new kind of magic square built up from monominoes each containing 1 to 16 dots. In this blog I will present you another form of magic square  containing varinoes magic as its elements. A varino is a single dot monomino with the the color of the dot and the the color of the background are varied. What are the puzzles around varinoes ?

Varino Magic Squares

If we have 4x4 square, can we distribute all possible 4-colors varinoes such that the dot colors and its background colors are not repeated in the same row, column and diagonal?
The following is one of the possible solutions:



Well, since every row column and diagonal has 4 different colored square and 4 different colored dots, they have the same set colored squares and the same set of colored dots, we will called the set pair as the magic set pair.

So, we have generalized the numeric magic square to a general magic square by replacing number with any similarly type element, and replacing the numeric addition with the set union. In this case the elements is the varinos.

Relation to Numeric Magic Square

In fact, the varino magic square is equivalent to numeric magic square, as it will be seen if we proceed the following replacement procedure:
If we replace
•    the red square by the numeral ‘0’ and the red dot by the numeral ‘0’
•    the blue square by the numeral ‘1’ and the blue dot by the numeral ‘1’
•    the green square by the numeral ‘2’and the green dot  by the numeral ‘2’
•    the yellow square by numeral ‘3’ and the yellow dot by the numeral ‘3’.
then we get the following numerical magic square

02	11	23	30
33	20	12	01
10	03	31	22
21	32	00	13

with the magic number 66 as the sum of all four numbers in any row, column and diagonal.
This numerical magic square can be transformed to the following numerical magic square

  2    5   11    12
15    8    6       1
  4    3   13    10
  9  14     0      7
with magic number 30, if we read the previous square as quaternary number base 4.
If we add 1 to every number in the last magic square, then we get the ordinary magic square of order 4

  3    6    12    13
16    9      7      2
  5    4    14    11
10  15      1     8

which isomorphic to the monominoes in the last blog

Afternote

In the next blog we will replace the varinoes with the combinoes which symbolize subsets of a set to get the combino magic square. This is another generalized magic square. Hopefully you will enjoy it as a mathematical recreation.

Unomino Magic Square

Unomino Magic Square
Armahedi Mahzar (c) 2015

A domino is two connected squares, each containing many dots or none. A unomino  is a single square containing dots like dominos. We can rearrange the little black square places so each column, row and diagonal is containing exactly the same numbers of dots to get


The unonimo magic square is isomorphic to the 3x3 magic square known as Lo Shu was discovered thousand years ago in the back of mythical turtle by Fuh-Shi,  the mythical founder of Chinese civilisation in around 2400 BC.


Before they invented the zero numeral, the Arabs used alphabets as the written symbols of numbers. Here is the 3×3 Magic Square



In fact there are bigger and bigger Magic Squares.

For example, the earliest 4X4 Magic Square is discovered in Khajuraho India dating
from the eleventh or twelfth century.



Later, another 4X4 Magic Square  was found in Albert Dürer’s engraving ” Melencolia”, where the date of its creation, 1514 AD. See it under the bell.



The previous nine monominoes is only part of larger set of monominoes containing dots from 1 up to 16. These are the sixteens monominos arranged is 4x4 square

We can rearrange the unominoes such that each column, row, diagonal and little 2×2 square is containing exactly the same number of dots. For example the following is the unomino magic square of order 4.
 


This magic square  is wonderful. Because all the collumn. rows and diagonals are containing 34 dots. The 2×2 center square is also containing 34 dots. The dots in each corner 2×2 squares are 34.

This is only one solution of the Puzzle. The French mathematician Frenicle de Bessy   in 1693 enumerated the number of all possible 4×4 Magic Square and get the number  880.