This document presents new general methods for constructing doubly even magic squares of order n=4k, where k is a positive integer. For each order n, the methods generate (n^2 - 2n/4)n/2 new magic squares that were previously unknown. An auxiliary matrix L of order n is defined, from which the new magic squares can be derived through a series of row and column swaps. Several examples of the auxiliary matrix L and resultant magic squares are provided for orders n=4 and n=8.

1. (), Vol. *, Issue * (20**); L. de O. Miranda, L.B.B.Miranda
Sequences of New Methods of Construction of Doubly
1
Even Magic Squares
2
Lohans de Oliveira Miranda1
, Lossian Barbosa Bacelar Miranda2
3
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 sequences of new methods of building doubly even magic squares. For
7
every n = 4k we build
n
2 − 2
n
4
n/2
new magic squares hitherto unknown.
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Keywords: arithmetic progressions, doubly even magic squares, parity.
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1 Introduction
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Here we present new general methods which builds, for each order, new types of magic squares hitherto
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unknown. A magic square of order n (or normal magic square) is a square matrix formed by the numbers
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1, 2, 3, ..., n2
and such that the sum of the numbers of each row, each column and each of the two diagonals
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is equal to cn = n3
+n
2 . We call cn of magic constant. The magic square is non-normal when the sum of
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the numbers in lines, columns and diagonals are all the same, however, not equal to cn = n3
+n
2 or the set
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of numbers that form it is not In
= {1, 2, 3, ..., n}. We call the aforementioned sums of totals. If n = 4k, k
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positive natural number, the magic square is of type 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
=
23
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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26
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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
4
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. 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. Sequences of New Methods of Construction of Doubly Even 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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3 Examples: auxiliary matrix, procedures i-ii and magic squares
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Order n = 4. L =




1 15 3 13
9 7 11 5
8 10 6 12
16 2 14 4



,




9 15 3 5
1 7 11 13
16 10 6 4
8 2 14 12



.
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52
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Order n = 8. L =












1 63 3 61 5 59 7 57
49 15 51 13 53 11 55 9
17 47 19 45 21 43 23 41
33 31 35 29 37 27 39 25
32 34 30 36 28 38 26 40
48 18 46 20 44 22 42 24
16 50 14 52 12 54 10 56
64 2 62 4 60 6 58 8












,












49 63 3 61 5 59 7 9
1 15 51 13 53 11 55 57
17 47 35 45 21 27 23 41
33 31 19 29 37 43 39 25
32 34 46 36 28 22 26 40
48 18 30 20 44 38 42 24
64 50 14 52 12 54 10 8
16 2 62 4 60 6 58 56












,
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55 











49 63 4 62 6 60 7 9
1 15 52 14 54 12 55 57
18 48 35 45 22 27 24 42
34 32 19 29 37 43 39 25
31 33 46 36 28 21 26 40
47 17 30 20 44 38 41 23
64 50 13 51 11 53 10 8
16 2 61 3 59 5 58 56












.
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4 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 not obtain infinite methods for any fixed n order.
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What we get are several sequences of methods which are valid only after a certain value of n. Note that for
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formula
n
2 − 2
n
4
n/2
to be true for n = 4, we must have (−1)! = 1.
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References
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