Here we have established definition of construction methods of magic squares and we prove the existence of infinite construction methods of doubly even magic squares.

1. (), Vol. *, Issue * (20**); L. de O. Miranda, L.B.B.Miranda
Functions and Methods of Construction of Magic
1
Squares
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Lohans de Oliveira Miranda1
, Lossian Barbosa Bacelar Miranda2
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1 Unisul University, Brazil
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2 IFPI, Brazil
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Correspondence to: Lossian Barbosa Bacelar Miranda, Email: lossianm@gmail.com
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Abstract: Here we have established definition of construction methods of magic squares and we prove the
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existence of infinite construction methods of doubly even magic squares.
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Keywords: arithmetic progressions, doubly even magic squares, functions, parity.
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1 Introduction
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Here we present definition of construction methods of magic squares. A magic square of order n (or normal
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magic square) is a square matrix formed by the numbers 1, 2, 3, ..., n2
and such that the sum of the numbers
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of each row, each column and each of the two diagonals is equal to cn = n3
+n
2 . We call cn of magic constant.
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The magic square is non-normal when the sum of the numbers in lines, columns and diagonals are all the
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same, however, not equal to cn
= n3
+n
2 or the set of numbers that form it is not In
= {1, 2, 3, ..., n}. We
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call the aforementioned sums of totals. If n = 4k, k positive natural number, the magic square is of type
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doubly even magic square.
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2 Main results
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Definition 1 (Auxiliary matrix). Consider the square matrix L = (lij)i,j=1,2,3,...,n
of n = 4k order, k ∈ N∗
,
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given by
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L =
A B
C D
=
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24 



(2r − 2) n + (2s − 1) n2
− (2r − 2) n − (2s − 1)
n2
− 2rn + (2s − 1) 2rn − (2s − 1)
(2r − 1) n − (2s − 1) n2
− (2r − 1) n + (2s − 1)
n2
− (2r − 1) n − (2s − 1) (2r − 1) n + (2s − 1)
2rn − 2 (s − 1) n2
− 2rn + 2s
n2
− (2r − 2) n − 2 (s − 1) (2r − 2) n + 2s
(2r − 1) n + 2s n
2
− (2r − 1) n − 2 (s − 1)
n2
− (2r − 1) n + 2s (2r − 1) n − 2 (s − 1)




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A, B, C and D are square matrices of order n/2, each in blocks of order 2, with r, s ∈ In
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as follows:
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1) in A, r grows from top to bottom and, s, grows from left to right; 2) in B, r grows from top to bottom
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and s grows from right to left; 3) in C, r grows from the bottom up and s, grows from left to right; 4) in
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D, r grows from the bottom up and s grows from right to left.
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Remark 1. A direct inspection says that: a) The sum of the elements of any column of L is equal to
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Cn; b) The sum of the elements of any of the first n/2 lines of L is equal to n3
/2; c) The sum of the ele-
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ments of any of the last n/2 lines of L is equal to Cn + n/2; d) The sum of the elements of main diagonal
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is n3
/4 + n/2; e) The sum of the elements of secondary diagonal is 3n3
/4 + n/2.
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Proposition 1 (MIRANDA, 2021b). From L can be built
n
2 − 2
n
4
n/2
doubly even magic squares.
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Proof. Let us, in L, do the following procedures: i) swap l2u−1,2u−1 with l2u,2u−1 and l2u−1,n+2−2u with
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l2u,n+2−2u when 1 6 u 6
n
4
; ii) swap l2u−1,2u−1 with l2u−2,2u−1 and l2u−1,n+2−2u with l2u−2,n+2−2u when
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n
4
u 6
n
2
. Note that swaps made with elements from any of the lines are compensated in pairs, so
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1

2. Functions and Methods of Construction of Magic Squares
that the sum of the elements of the lines remains unchanged. These swaps also do not change the sum
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of the elements in any column. However, a simple direct calculation shows that the set of swaps adds
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n3
/4 units to the main diagonal and removes −n3
/4 units from the secondary diagonal. Consequently, the
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sum of the elements of both diagonals becomes Cn. An inspection on matrices A, B, C and D shows that
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lv,2t−1 = ln+1−v,2t + 1, lv,2t = ln+1−v,2t−1 + 1; ∀v ∈ In, ∀t ∈ In
2
. This implies that we can transfer n/2 units
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from the line of v order (v n/2) to the line of n + 1 − v order in
n
2 − 2
n
4
ways, since in just two pairs
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of adjacent columns we cannot do this transfer due to the aforementioned i-ii procedures. Since there are
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n/2 pairs of lines and the transfers are independent, then we will have a total of
n
2 − 2
n
4
n/2
possibilities.
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Definition 2. Let S = {n1, n2, n3, ...} be a subsequence of N∗
{2}. We call of magic square construction
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method associated with S to any function f : S → M; ni 7→ Mni
, with M being the set of all magic squares
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and Mni
a magic square of order ni. We call f : {n1, n2, n3, ..., nk} → M truncation of f in order k. Two
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methods are the same if the functions that define them are the same.
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Proposition 2. Consider that a successive procedure defined in S = {n1, n2, n3, ...} produces, in inde-
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pendent ways, g(ni) 0 magic squares for the order ni, ∀i ∈ N∗
. Then: i) can be produced, from this
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successive procedure, magic squares construction methods associated with S; ii) there are
Qk
i=1 g(ni) trun-
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cations of these methods in order k; iii) If g(ni) 2 for an infinite amount of i values, there will be infinite
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magic squares construction methods associated with S.
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Proof. i-ii) To build the first method, choose a magic square for each of the orders (different of 2). The
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first method will be determined. To build the second method, choose a magic square of order any not yet
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chosen in the construction of the first method. Then, add to this magic square others, in all orders, not
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yet chosen in the first method. If for some orders it is not possible to choose new magic squares, repeat the
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ones already chosen in the first method. This second construction method of magic squares associated with
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S will be different from the first method. To build the third method, redo the previous procedure by choos-
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ing magic squares not yet chosen in the methods already built and so on, to build more methods. In this
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way, all methods will be, two by two, different. As the choices are independent, the fundamental principle
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of counting can be applied. iii) It is an immediate consequence of the previous item with the hypothesis
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g(ni) 2.
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Observation 1. The result of Proposition 1 together with the Proposition 2 indicates that the proce-
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dure that generates
n
2 − 2
n
4
n/2
magic squares for each ni produces infinite construction methods of magic
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squares.
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3 Discussion
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The Proposition 1 makes the doubly even magic squares to be demonstrably abundant among the three main
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types of magic squares. It should be noted that we do obtain infinite construction methods of doubly even
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magic squares. The Definition 2 does not change the usual concept of construction method of magic squares
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that has been known since ancient times. Note that we can combine the various methods known since
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ancient times to generate infinite other methods. However, these combinations, as a rule, will not generate
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new squares in addition to those already known. We can also build infinite construction methods of doubly
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even magic squares using the results obtained in (MIRANDA, 2021a).
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References
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[1] HOLGER DANIELSSON. Magische Quadrate, Version 2.03 vom 04.10.2020a. Available on
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https://www.magic-squares.info/docs/magische-quadrate.pdf. Access on 08/30/2020.
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[3] MIRANDA, Lohans de O. and MIRANDA, Lossian B. B. Lohans’ Magic Squares and the Gaussian
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Elimination Method. Journal of Nepal Mathematical Society (JNMS), Volume 3, Issue 1, Year-2020a.
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Magic Square and New Methods for Doubly Even Magic Squares, 2020b. Available on
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[8] MIRANDA, Lohans de O. and MIRANDA, Lossian B. B. Establishing In-
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[9] MIRANDA, Lohans de O. and MIRANDA, Lossian B. B. Sequences of New Methods of Construction
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