On the Existence of a Normal Trimagic Square of Order 16 n

. Te study of magic squares has a long history, and magic squares have been applied to many mathematical felds. In this paper, we give a complete solution to the existence of normal trimagic squares of all orders 16 n . In particular, we obtain a unifed solution for the normal trimagic square of order 16 n for n > 3 by means of set partitions, semibimagic squares, Latin squares, and new product construction. Since there exist normal trimagic squares of orders 16, 32, and 48, we prove that there exists a normal trimagic square of order 16 n for every positive integer n .


Introduction
Magic squares are among the oldest known combinatorial designs and have been applied to many felds of mathematics [1][2][3].Teir origin can be found in the frst-century book Da-Dai Liji in China [4].A magic square of order n, denoted by MS (n), is an n × n matrix consisting of integers such that the sum of the entries in each row, each column, and each of two diagonals is the same number called the magic sum.A magic square of order n is normal if its entries are n 2 consecutive integers, say 0, 1, . . ., n 2 − 1.
During this long period of time, many subclasses have been proposed, such as bimagic squares and trimagic squares.Let t be a positive integer.A magic square of order n is a tmultimagic square, denoted by MS (n, t), if it remains magic when all its entries are replaced by its d-th powers for d ∈ 1, . . ., t { }.Usually, a 2-multimagic square is called a bimagic square and a 3-multimagic square is called a trimagic square.A lot of work has been done on normal magic squares and normal multimagic squares; for more details, the interested reader may refer to [5][6][7][8][9][10][11] and the references therein.
In this paper, we frst present some related preliminaries (Section 2), then prove our main results and provide a brief discussion (Section 3), and fnally give conclusions about the work (Section 4).

Preliminaries
Tis section presents some related preliminaries which are used in this paper.
Let J n be an n × n matrix with the entries being all 1's and let For integers a and b with a ≤ b and a ≡ b (mod 2), let [a, b] 2 denote the set a, a + 2, . . ., { b − 2, b}.For an n × n array A, the rows and columns of A are indexed by I n .For any n × n array A, we denote by S A the set of the entries of A. Usually, the (i, j) entry of an array A is denoted by a i,j .An MS (n, For an integer set S, let S � − x: x ∈ S { }.Let n be an integer greater than 3 and S n � (1/2) 4n − 1 k�0 (2k + 1) 2 .Te pair of P and Q is called a bimagic subset pair, denoted by BSP (n), if the following three conditions are satisfed: Using a computer search, we obtain the following bimagic subset pairs.
then it is easy to check that the pair of P n and Q n is a BSP (n) for n ∈ 4, 5, 6, 7 { }.Semimagic squares are a generalization of magic squares.A semimagic square of order n, denoted by SMS (n), is an n × n matrix consisting of integers such that the sum of the entries in each row and each column is the same number called the magic sum.Tus every magic square is a semimagic square, but not the converse.Let t be a positive integer.A semimagic square of order n is a semi t-multimagic square, denoted by SMS (n, t), if it remains semimagic when all its entries are replaced by its d-th powers for d ∈ 1, . . ., t { }.For results of semimultimagic squares, we refer the reader to [18] and the references therein.Usually, a semi 2-multimagic square is called a semi-bimagic square.An SMS (n, t) is also called a t-multimagic rectangle in [18].A semibimagic square especially is called a bimagic square rectangle.Terefore, semimultimagic squares defned in this paper are special subclasses of multimagic rectangles defned in [18].
Based on the literature [6], we write A * d � (a d i,j ) for any positive integer d.We call the matrix A consisting of distinct integers self-complementary if S − A � S A .
Let A, B, C, and D be integer matrices of order 4n.We call the tuple (A, B, C, D) an extendable tuple of order 4n, denoted by ET (4n), if the following conditions are satisfed: (E 2 ) A, B, C, and D are self-complementary and each of their diagonals consists of opposite numbers; (E 3 ) A, B, C, and D are MS (4n) s with magic sum 0, B * 2 and C * 2 are SMS (4n) s with magic sum 2(64n 2 +1)S n , and A * 2 and D * 2 are MS (4n) s with magic sum 2(64n 2 +1)S n .
Te following defnition is from [11].Let T be an n-set.A diagonal Latin square of order n over T, denoted by DLS (n), is an n × n array such that the set of the entries in each row, each column, and each of two diagonals is T. Two DLS (n) s are called orthogonal if each symbol in the frst square meets each symbol in the second square exactly once when they are superposed.In this paper, we need the following.

Lemma 2 (see Abel et al. [19]). Two orthogonal diagonal Latin squares of order n exist if and only if
Using the literature [20], we give the following defnition.An n × n array W with the entries in an n-set T is a balanced square if each element of T appears n times in W. Two balanced squares are called orthogonal if each symbol in the frst square meets each symbol in the second square exactly once when they are superposed.
For an integer set T and an integer x, denote x + t: t ∈ T { } by x + T.

Results and Discussion
In this section, frst we shall show that a normal MS (n, t) can be obtained by constructing a quasi-normal MS (n, t), then construct bimagic subset pairs and semibimagic squares, next use new product construction to get a quasi-normal MS (16n, 3) for n > 3 and prove our main theorem, and fnally give a generalization.Terefore, we reduce the problem for the construction of a normal MS (n, t) to the problem for the construction of a quasi-normal MS (n, t).

Construction of Bimagic Subset Pairs.
In this section, we shall prove the following core result.For the induction step, let w be a positive integer and k � 4w + v and assume that P v (w) is true; that is, there exists a BSP (k), P k , and Q k , satisfying the following conditions: ( Let m � k + 4. We want to show that then P v (w + 1) is true; that is, there exists a BSP (m), P m , and Q m , satisfying the following conditions: ( Let Ten, it is easy to check that (5) Clearly, we have Journal of Mathematics By our induction hypothesis (B 2 ), we get By our induction hypothesis (B 3 ), we obtain Taking P m � P k ∪ P * and Q m � Q k ∪ Q * , from the induction hypotheses (B 1 ), (B 2 ), and (B 3 ), we see that the conditions (B 4 ), (B 5 ), and (B 6 ) hold; in other words, the statement P v (w + 1) is true.Tus, assuming P v (w) is true, it follows that P v (w + 1) is true.For v ∈ I 4 , by induction, P v (u) is true for u ≥ 1.In summary, there exists a BSP (n) for n ≥ 4.

Construction of Extendable Tuples (Semibimagic Squares).
In this section, we shall construct the building blocks forming quasi-normal trimagic squares by taking advantage of extendable tuples consisting of semibimagic squares.We shall take advantage of construction methods for quasinormal MS (4n, 2) s introduced by Pan and Huang [21], that is, combine the existence of a pair of orthogonal diagonal Latin squares of order 2n such that n ≥ 4 with a BSP (n), to construct such an ET (4n).Now, we state the following.

Lemma 5.
Tere exists an extendable tuple of order 4n for n ≥ 4.
Proof.Let k * denote 4n − 1− k for k ∈ I 4n and let δ k � (− 1) ⌊k/n⌋ for k ∈ I 2n .Let the pair of P and Q be a BSP (n) and (L, R) ∈ (P, P), Suppose that E and F are orthogonal diagonal Latin squares over L and R, respectively.Write E � (e i,j ) and − e i,j − e i,j When (L, R) � (P, P), denote G + 8nH by A; when Next, we shall prove that the square tuple (A, B, C, D) is an ET (4n).
First, we prove that A, B, C, and D satisfy (E 1 ).Since E and F are orthogonal, αE and βF are orthogonal, where Terefore, we obtain 4 Journal of Mathematics Next, we prove that A, B, C, and D satisfy (E 2 ).Since we see at once that W consists of (4n) 2 distinct integers.From ( 9), for i, j ∈ I 2n , we have w i,j � g i,j + 8nh i,j � e i,j + 8nδ j f i,j , � − e i,j + 8nδ j f i,j . ( Hence, the square W is self-complementary and each of two diagonals consists of opposite numbers.Since W ∈ A, B, C, D { }, we prove that A, B, C, and D are selfcomplementary and each of their diagonals consists of opposite numbers.Now, we prove that A, B, C, and D satisfy (E 3 ).From (9), for i ∈ I 2n , we have Similarly, for j ∈ I 2n , we have Noting that each of two diagonals of W consists of opposite numbers, we know that W is an MS (4n) with magic sum 0. Terefore, we prove that A, B, C, and D are four MS (4n) s with magic sum 0.

Journal of Mathematics
Further, for L � R, we shall show that the sum of the entries in each of two diagonals of W * 2 is 2(64n 2 + 1)S n .We have Hence, A * 2 and D * 2 are MS (4n) s with magic sum 2(64n 2 + 1)S n .□

New Product Construction and Proof of Main Teorem.
In this section, using an ET (4n) and a new product construction, we shall give a unifed construction of all quasinormal MS (16n, 3) s for n ≥ 4 and prove our main theorem.Lemma 6.If there exists an ET (4n), then there exists a quasi-normal MS (16n, 3).(A, B, C, D) be an ET (4n), and let

Proof. Let
where E ⊗ J 4n is the Kronecker product of E and J 4n , that is, E ⊗ J 4n � (e i,j J 4n ).Set G � (g i,j ) (i, j ∈ I 16n ).Write F u,v � (f (u,v) r,s ), and then we have In the following, we prove that G is a quasi-normal MS (16n, 3).
First, we shall prove that G is quasi-normal.Noting that S − A � S A , S − B � S B , S − C � S C , and S − D � S D , we get a special array F * as follows: Terefore, F * is an array of order 4 over the symbol set S A , S B , S C , S D  , that is, S F * .Obviously, E and F * are both balanced squares, and E and F * are orthogonal.

6
Journal of Mathematics Terefore, noting that It follows that which means that G is quasi-normal.Now, we show that G is a magic square.Noting that E is a DLS(4) over − 3, − 1, 1, 3 { } and ±A, ±B, ±C and ±D are magic squares with magic sum 0, for i ∈ I 16n , we have Similarly, one can prove that Next, we show that G * 2 is a magic square.Let T n � 2(64n 2 +1)S n .Noting that E * 2 is a magic square with magic sum 20, matrices (± A) * 2 and (± D) * 2 are magic squares with magic sum T n , and (± B) * 2 and (± C) * 2 are semimagic squares with magic sum T n , for i ∈ I 16n , we obtain Write T � (8n) 4 × 80n + 4T n .Similarly, one can prove that Terefore, G * 2 is a magic square; that is, G is a quasinormal bimagic square.
Finally, we show that G * 3 is a magic square.Noting that E * 3 is a magic square with magic sum 0 and each column of F consists of opposite numbers, for i ∈ I 16n , we obtain Journal of Mathematics Similarly, one can prove that Terefore, G * 3 is magic; that is, G is a quasi-normal trimagic square.
We are now in a position to prove the main result.□ Theorem .Tere exists a normal MS (16n, 3) for all positive integers n.

Conclusion
In the paper, we reduce the problem for the construction of a normal t-multimagic square of order n to the problem for the construction of a quasi-normal t-multimagic square of order n and give a complete solution to the existence of normal trimagic squares of all orders 16n with the help of many other combinatorial confgures, such as set partitions, bimagic subset pairs, semibimagic squares, Latin squares, and classical product construction.More precisely, we prove that there exists a normal trimagic square of order 16n for all positive integers n.Meanwhile, the concept of the extendable tuple and the conclusion of Lemma 6 are generalized to the case of (2t + 1)-th power.

Lemma 4 .
Tere exists a BSP (n) for n ≥ 4.Proof.Let v ∈ I 4 and u be a positive integer.Now, we prove the statement P v (u).Tere exists a BSP (4u + v) for u ≥ 1 by induction.Te base case P v (1) is obviously true by Example 1.