Semi-magic dihedral squares

Semi-magic dihedral squares Sylwia Cic hacz 1 , Dalib or F roncek 1 , 2 1 A GH Univ ersity of Krako w, Poland, cichacz@agh.edu.pl 2 Univ ersity of Minnesota Duluth, U.S.A., dalib or@d.umn.edu Abstract Let Γ b e a group of order n 2 and S M S Γ ( n ) = ( a i,j ) n × n b e an n × n arra y whose en tries are all distinct elemen ts of Γ. If there exists an element µ ∈ Γ such that for every row i , there exists an ordering of elements such that a i,j 1 a i,j 2 . . . a i,j n − 1 a i,j n = µ and for ev ery column j there exists an ordering of elements such that a i 1 ,j a i 2 ,j . . . a i m − 1 ,j a i m ,j = µ, then S M S Γ ( n ) is called a Γ -semi-magic squar e of side n and µ is called a magic c onstant . W e pro vide a complete c haracterization of semi-magic squares of side n whose en tries belong to a dihedral group D k . Moreo ver, w e show that in our constructions a single semi-magic square may admit t wo distinct magic constan ts, dep ending on the order in which the pro ducts are computed. Keyw ords: Magic squares, magic rectangles, dihedral group 2000 Mathematics Sub ject Classification: 05B15 1 In tro duction Magic squares are one of the oldest mathematical structures, rep ortedly dating bac k to the 4-th century BCE. A magic squar e of order n is an n × n array with en tries 1 , 2 , . . . , n 2 suc h that the sum of eac h ro w, column, and the main forw ard and backw ard diagonal i s the same magic c onstant c = n ( n 2 + 1) / 2. When we only require the row and column sum to b e equal and disregard the diagonals, w e sp eak ab out a semi-magic squar e. F or an ov erview, see, e.g., Chapter 6 Section 33 in [4]. A complete c haracterization is w ell kno wn. Theorem 1.1 ([4]) . Ther e exists a magic squar e of side n if and only if n > 2 . A generalization of a magic square is an m × n magic r e ctangle with entries 1 , 2 , . . . , mn where all ro w sums are equal to the r ow c onstant r = n ( mn + 1) / 2 all column sums are equal to the c olumn c onstant c = m ( mn + 1) / 2. A semi- magic square is then an n × n magic rectangle. Magic squares and rectangles can b e generalized in many different wa ys. F or instance, we may require that the 1 en tries are elements of an Abelian group (see [1, 2, 3, 5, 7, 8, 9]). In con trast, the case of non-Ab elian groups has so far b een considered only in [6]. Namely , the second author recently tac kled the problem of dihedral groups and constructed D 2 n 2 -semimagic rectangles of side n ov er dihedral groups D 2 n 2 of order 4 n 2 for an y n ≡ 0 (mo d 4) [6]. In this pap er, w e extend this result to all admissible v alues of n , using simpler construction metho ds. F urthermore, we show that in our constructions, a given s emi-magic square may admit tw o distinct magic constan ts, dep ending on the order in whic h the pro ducts are taken. Disclaimer Some p arts of this and the fol lowing se ction may b e similar or identic al to c orr esp onding p arts of p ap er [6] by the se c ond author on similar topic. 2 Definitions and necessary conditions It is common to refer to an n × n magic square as a square of or der n . T o av oid confusion b etw een the order of the group Γ and the square, we reserve the word “order” to the group and sp eak ab out magic rectangle of side n . W e use the notation [ a, b ] for the set { a, a + 1 , . . . , b } of consecutiv e integers. When a = 0, w e ma y use just [ b ]. Definition 2. 1. Let Γ b e a group of order mn and M R Γ ( m, n ) = ( a i,j ) an m × n array whose entries are all elements of Γ. If for every row i there exists an ordering of elements such that a i,j 1 a i,j 2 . . . a i,j n − 1 a i,j n = ρ and for every column j there exists an ordering of elements such that a i 1 ,j a i 2 ,j . . . a i m − 1 ,j a i m ,j = σ, then M R Γ ( m, n ) is called Γ -magic r e ctangle. If for every row and column the ordering is a i, 1 a i, 2 . . . a i,n − 1 a i,n = ρ and a m,j a m − 1 ,j . . . a 2 ,j a 1 ,j = σ, then M R Γ ( m, n ) is line arly Γ -magic. If for every ro w i there exists j (not necessarily the same for all i ) suc h that a i,j +1 a i,j +2 . . . a i,n a i, 1 . . . a i,j − 1 a i,j = ρ and for every column j there exists i such that a i − 1 ,j a i − 2 ,j . . . a m,j a 1 ,j . . . a i +1 ,j a i,j = σ, 2 then we call the ordering cir cular and M R Γ ( m, n ) cir cularly Γ -magic . Finally , if m = 2 s, n = 2 t and for every ro w i there exists j ∈ [1 , t ] and j ′ ∈ [ t + 1 , n ] (not necessarily the same for all i ) such that a i,j +1 a i,j +2 . . . a i,s a i, 1 . . . a i,j − 1 a i,j = ρ 1 and a i,j ′ +1 a i, ′ +2 . . . a i,m a i,s +1 . . . a i,j ′ − 1 a i,j ′ = ρ 2 and for every column j there exists i ∈ [1 , s ] and i ′ ∈ [ s + 1 , m ] such that a i − 1 ,j a i − 2 ,j . . . a 1 ,j a n,j . . . a i +1 ,j a i,j = σ 1 , and a i ′ − 1 ,j a i ′ − 2 ,j . . . a 1 ,j a n,j . . . a i ′ +1 ,j a i ′ ,j = σ 2 , then we call the ordering semi-cir cular and M R Γ ( m, n ) semi-cir cularly Γ -magic . Definition 2.2. Let Γ b e a group of order n 2 and M R Γ ( n, n ) a Γ-magic rect- angle. If the row and column pro ducts are equal, that is, ρ = σ , then we call M R Γ ( n, n ) a Γ -semi-magic squar e and if moreov er the pro ducts of b oth the main and backw ard main diagonals are equal to ρ = σ , then M R Γ ( n, n ) is called a Γ -magic squar e . The notions of line arly, cir cularly and semi-cir cularly Γ -magic and Γ -semi- magic squares are defined as in Definition 2.1. Observ e that in the case where Γ is an Ab elian group, the order of ele- men ts in each pro duct is irrelev an t, as the group op eration is commutativ e. Consequen tly , the conditions on the orderings in the definitions ab ov e are au- tomatically satisfied. The dihedral group D k of order 2 k (sometimes also denoted by D 2 k ) is the group consisting of k rotations r i and k reflections s i , where the rotations form a cyclic group of order k and and each reflection generates a subgroup of order 2. More formal definition is b elo w. Definition 2.3. The dihe dr al gr oup D k of order 2 k where k ≥ 3 is defined on the set of elemen ts { r 0 , r 1 , . . . , r k − 1 , s 0 , s 1 , . . . , s k − 1 } where r 0 = e, r 1 = r, r i = r i , s 0 = s , s i = r i s , s 2 i = e and r i s = sr − i for i = 0 , 1 , . . . , k − 1. The elements r i are called r otations , and s i are called r efle ctions . An imp ortan t prop ert y of D k will b e used in our constructions. If follows directly from the definition. Prop osition 2.4. In any dihe dr al gr oup D k , we have sr i s = r − i for every i = 0 , 1 , . . . , k − 1 . 3 When w e ha v e a Γ-magic or semi-magic square where Γ = D k , the dihedral group on 2 k elements, we sp eak ab out a dihe dr al magic or semi-magic squar e . W e b egin by stating the ob vious necessary conditions. Since every dihedral group is of even order, a magic square M S D k of o dd order cannot exist. The follo wing necessary condition follows from straightforw ard observ ations made in [6]. Theorem 2.5 ([6]) . If a dihe dr al semi-magic squar e S M S D k ( n ) exists, then b oth n and k must b e even. 3 Preliminaries By Theorem 2.5, the necessary condition for the existence of S M S D k ( n ) is that k and n are even, thus n = 4 m 2 and k = 2 m 2 for some p ositive integer m > 2. Theorem 1.1 implies the existence a magic square ˜ M ( m ) of side m > 2 with the magic constan t ˜ µ . Using the square ˜ M ( m ) = ( ˜ m i,j ) m × m w e build three p ower squar es of size m × m : E ( m ), O ( m ) and T ( m ) as follo ws. W e will later use them to construct S M S D 2 m 2 (2 m ). Definition 3.1. Let ˜ M ( m ) b e a magic square with entries ˜ m i,j and magic constan t ˜ µ . Define p ower squar es E ( m ) , O ( m ) and T ( m ) with entries e i,j , o i,j and t i,j , resp ectively , for i, j ∈ [0 , m − 1] as follows: • e i,j = 2 ˜ m i,j (mo d 2 m 2 ) • o i,j = 2 ˜ m i,j +1 (mo d 2 m 2 ) • t i,j = − 2 ˜ m i,j + 1 (mo d 2 m 2 ) when m is even • t i,j = − 2 ˜ m i,j + m − 2 (mo d 2 m 2 ) when m is o dd W e notice that in b oth cases t i,j is o dd. The sums in rows and columns of the squares are constant, as can b e easily v erified. W e calculate them again mo dulo 2 m 2 . Observ ation 3.2. The r ow and c olumn sums in the p ower squar es E ( m ) , O ( m ) , and T ( m ) ar e • ˜ e = P m − 1 j =0 e i,j = P m − 1 i =0 2 ˜ m i,j = 2 ˜ µ (mo d 2 m 2 ) • ˜ o = P m − 1 j =0 o i,j = P m − 1 i =0 (2 ˜ m i,j + 1) = (2 ˜ µ + m ) (mo d 2 m 2 ) • ˜ t = P m − 1 j =0 t i,j = P m − 1 i =0 ( − 2 ˜ m i,j + 1) = ( − 2 ˜ µ + m ) (mo d 2 m 2 ) for m even • ˜ t = P m − 1 j =0 t i,j = P m − 1 i =0 ( − 2 ˜ m i,j + m − 2) = ( − 2 ˜ µ + m 2 − 2 m ) (mo d 2 m 2 ) for m o dd 4 Example 3.3. In Figure 1 we sho w the construction of p ow er squares E (4), O (4) and T (4). 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 (a) ˜ M (4), ˜ µ = 34 0 4 6 26 10 22 20 16 18 14 12 24 8 28 30 2 (b) E (4), ˜ e = 4 1 5 7 27 11 23 21 17 19 15 13 25 9 29 31 3 (c) O (4), ˜ o = 8 1 29 27 7 23 11 13 17 15 19 21 9 25 5 3 31 (d) T (4), ˜ t = 0 Figure 1: Example of E (4), O (4) and T (4) Example 3.4. In Figure 2 we sho w the construction of p ow er squares E (5), O (5) and T (5). 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 (a) ˜ M (5), ˜ µ = 65 34 48 2 16 30 46 10 14 28 32 8 12 26 40 44 20 24 38 42 6 22 36 0 4 18 (b) E (5), ˜ e = 30 35 49 3 17 31 47 11 15 29 33 9 13 27 41 45 21 25 39 43 7 23 37 1 5 19 (c) O (5), ˜ o = 35 19 5 1 37 23 7 43 39 25 21 45 41 27 13 9 33 29 15 11 47 31 17 3 49 35 (d) T (5), ˜ t = 35 Figure 2: Example of E (5), O (5) and T (5) 5 Rather than using the group elements and entries and p erforming the group op eration, w e simply use the exp onents in elements r i = r i and s i = r i s . Then, when in our square w e w ould p erform the pro duct (read from right to left) . . . r i r j . . . , w e instead just add the exp onents . . . i + j . . . . F or reflections, we recall that from Prop osition 2.4 it follows that ( r i s )( r j s ) = r i ( sr j s ) = r i r − j and we use just the term i − j . W e will build M S Γ ( n ) from partial squares Q uv ( m ) = ( q uv i,j ) m × m for u, v ∈ { 1 , 2 } . Because the partial squares are alw ays identified by the sup erscript uv , w e b eliev e that no confusion will arise when w e drop the square sides and write simply Q uv instead of Q uv ( m ). The row sums in ro w i will b e ρ uv i in Q uv and it is calculated according to the order: ρ uv i = q uv i,i q uv i,i +1 . . . q uv i,m − 1 q uv i, 0 q uv i, 1 . . . q uv i,i − 1 , where the subscripts are taken mo dulo m . In other words, w e alwa ys start with the en try just before the square diagonal and pro ceed to the left cyclically until we stop at the diagonal. The column sums in column j will b e σ uv j and it is calculated according to the order: σ uv j = q uv j,j q uv j − 1 ,j . . . q uv 0 ,j q uv m − 1 ,j q uv m − 2 ,j . . . q uv j +1 ,j . Similarly as in the ro w pro duct, we alwa ys start with the entry b elow the square diagonal and pro ceed do wn cyclically un til w e arriv e at the diagonal. 4 Construction for m ev en W e present a general construction for k ≥ 8, using the partial squares Q uv . The squares Q 11 and Q 22 will consist of rotations and Q 12 and Q 21 of reflections. Construction 4.1. R otations W e construct first the squares Q 11 and Q 22 . Let E ( m ) = ( e i,j ) m × m and O ( m ) = ( o i,j ) m × m b e the p ow er squares defined in Section 3. Let q 11 i,j = ( r e i,j , i + j ev en , r o i,j , i + j odd , q 22 i,j = ( r e i,j +1 , i + j even , r o i,j +1 , i + j odd . Observ e that for any i, j ∈ [0 , m − 1] w e ha ve e i,l + o i,l +1 = 2 ˜ m i,l + 2 ˜ m i,l +1 + 1 , o l − 1 ,j + e l,j = 2 ˜ m l − 1 ,j + 1 + 2 ˜ m l,j , 6 where the subscripts are taken mo dulo m . Therefore ρ 11 i = r e i,i r o i,i +1 r e i,i +2 r o i,i +3 . . . r e i,i − 2 r o i,i − 1 = r 2 ˜ µ + m/ 2 , σ 11 j = r e j,j r o j − 1 ,j r e j − 2 ,j r o j − 3 ,j . . . r e j +2 ,j r o j +1 ,j = r 2 ˜ µ + m/ 2 , ρ 22 i = r e i,i +1 r o i,i +2 r e i,i +3 r o i,i +4 . . . r e i,i − 1 r o i,i = r 2 ˜ µ + m/ 2 , σ 22 j = r e j − 1 ,j r o j − 2 ,j r e j − 3 ,j r o j − 4 ,j . . . r e j +1 ,j r o j,j = r 2 ˜ µ + m/ 2 . W e show an example for m = 4 in Figure 3. r e 1 , 1 r o 1 , 2 r e 1 , 3 r o 1 , 4 r o 2 , 1 r e 2 , 2 r o 2 , 3 r e 2 , 4 r e 3 , 1 r o 3 , 2 r e 3 , 3 r o 3 , 4 r o 4 , 1 r e 4 , 2 r o 4 , 3 r e 4 , 4 = r 0 r 5 r 6 r 27 r 11 r 22 r 21 r 16 r 18 r 15 r 12 r 25 r 9 r 28 r 31 r 2 (a) Q 11 (4) r e 1 , 2 r o 1 , 3 r e 1 , 4 r o 1 , 1 r o 2 , 2 r e 2 , 3 r o 2 , 4 r e 2 , 1 r e 3 , 2 r o 3 , 3 r e 3 , 4 r o 3 , 1 r o 4 , 2 r e 4 , 3 r o 4 , 4 r e 4 , 1 = r 4 r 7 r 26 r 1 r 23 r 20 r 17 r 10 r 14 r 13 r 24 r 19 r 29 r 30 r 3 r 8 (b) Q 22 (4) Figure 3: Squares Q 11 and Q 22 Construction 4.2. R efle ctions W e construct now the squares Q 12 and Q 21 . Let E ( m ) = ( e i,j ) m × m , O ( m ) = ( o i,j ) m × m and T ( m ) = ( t i,j ) m × m b e the pow er squares defined in Section 3. Let q 12 i,j = ( r e i,j s, i + j ev en , r t i,j s, i + j odd , q 21 i,j = ( r e i,j +1 s, i + j ev en , r t i,j +1 s, i + j odd . Observ e that for any i, j, l ∈ [0 , m − 1] we hav e e i,l − t i,l +1 = 2 ˜ m i,l − ( − 2 ˜ m i,l +1 + 1) = 2 ˜ m i,l + 2 ˜ m i,l +1 − 1 , e l,j − t l − 1 ,j = 2 ˜ m l,j − ( − 2 ˜ m l − 1 ,j + 1) = 2 ˜ m l,j + 2 ˜ m l − 1 ,j − 1 , 7 where the subscripts are taken mo dulo m . Therefore, ρ 12 i = r e i,i sr t i,i +1 sr e i,i +2 sr t i,i +3 s . . . r e i,i − 2 sr t i,i − 1 s = r e i,i r − t i,i +1 r e i,i +2 r − t i,i +3 s . . . r e i,i − 2 r − t i,i − 1 s = r 2 ˜ µ − m/ 2 , σ 12 j = r e j,j sr t j − 1 ,j sr e j − 2 ,j sr t j − 3 ,j s . . . r e j +2 ,j sr t j +1 ,j s = r e j,j r − t j − 1 ,j r e j − 2 ,j r − t j − 3 ,j . . . r e j +2 ,j r t j +1 ,j = r 2 ˜ µ − m/ 2 , ρ 21 i = r e i,i +1 sr t i,i +2 sr e i,i +3 sr t i,i +4 s . . . r e i,i − 1 sr t i,i s = r e i,i +1 r − t i,i +2 r e i,i +3 r − t i,i +4 . . . r e i,i − 1 r − t i,i s = r 2 ˜ µ − m/ 2 , σ 21 j = r e j − 1 ,j sr t j − 2 ,j sr e j − 3 ,j sr t j − 4 ,j s . . . r e j +1 ,j sr t j,j s = r e j − 1 ,j r − t j − 2 ,j r e j − 3 ,j r − t j − 4 ,j . . . r e j +1 ,j r − t j,j = r 2 ˜ µ − m/ 2 . W e show an example for m = 4 in Figure 4. r e 1 , 1 s r t 1 , 2 s r e 1 , 3 s r t 1 , 4 s r t 2 , 1 s r e 2 , 2 s r t 2 , 3 s r e 2 , 4 s r e 3 , 1 s r t 3 , 2 s r e 3 , 3 s r t 3 , 4 s r t 4 , 1 s r e 4 , 2 s r t 4 , 3 s r e 4 , 4 s = r 0 s r 29 s r 6 s r 7 s r 23 s r 22 s r 13 s r 16 s r 18 s r 19 s r 12 s r 9 s r 25 s r 28 s r 3 s r 2 s (a) Q 12 (4) r e 1 , 2 s r t 1 , 3 s r e 1 , 4 s r t 1 , 1 s r t 2 , 2 s r e 2 , 3 s r t 2 , 4 s r e 2 , 1 s r e 3 , 2 s r t 3 , 3 s r e 3 , 4 s r t 3 , 1 s r t 4 , 2 s r e 4 , 3 s r t 4 , 4 s r e 4 , 1 s = r 4 s r 27 s r 26 s r 1 s r 11 s r 20 s r 17 s r 10 s r 14 s r 21 s r 24 s r 15 s r 5 s r 30 s r 31 s r 8 s (b) Q 21 (4) Figure 4: Squares Q 12 and Q 21 W e are now ready to state our first result. Theorem 4.3. Ther e exists a semi-cir cularly D 2 m 2 -semi-magic squar e Q (2 m ) for every even m , m ≥ 4 . Pr o of. Let Q u,v b e the four squares obtained by Constructions 4.1 and 4.2 W e will glue them to obtain a square Q as in Figure 5. 8 r 0 r 5 r 6 r 27 r 0 s r 29 s r 6 s r 7 s r 11 r 22 r 21 r 16 r 23 s r 22 s r 13 s r 16 s r 18 r 15 r 12 r 25 r 18 s r 19 s r 12 s r 9 s r 9 r 28 r 31 r 2 r 25 s r 28 s r 3 s r 2 s r 4 s r 27 s r 26 s r 1 s r 4 r 7 r 26 r 1 r 11 s r 20 s r 17 s r 10 s r 23 r 20 r 17 r 10 r 14 s r 21 s r 24 s r 15 s r 14 r 13 r 24 r 19 r 5 s r 30 s r 31 s r 8 s r 29 r 30 r 3 r 8 Figure 6: S M S D 32 (8) with the magic constant µ = r 8 Q 11 Q 12 Q 21 Q 22 Figure 5: A square Q = S M S D 2 m 2 (2 m ) Eac h ro w pro duct is now p erformed as ρ i = ( q u 1 i,i q u 1 i,i +1 . . . q u 1 i,m − 1 q u 1 i, 0 q u 1 i, 1 . . . q u 1 i,i − 1 ) ( q u 2 i,i q u 2 i,i +1 . . . q u 2 i,m − 1 q u 2 i, 0 q u 2 i, 1 . . . q u 2 i,i − 1 ) = ρ u 1 i ρ u 2 i = r 2 ˜ µ + m/ 2 r 2 ˜ µ − m/ 2 = r 4 ˜ µ and the column pro ducts as σ j = ( q 1 v j,j q 1 v j − 1 ,j . . . q 1 v 0 ,j q 1 v m − 1 ,j q 1 v m − 2 ,j . . . q 1 v j +1 ,j ) ( q 2 v j,j q 2 v j − 1 ,j . . . q 2 v 0 ,j q 2 v m − 1 ,j q 2 v m − 2 ,j . . . q 2 v j +1 ,j ) = σ 1 v i σ 2 v i = r 2 ˜ µ + m/ 2 r 2 ˜ µ − m/ 2 = r 4 ˜ µ , where the subscripts are taken mo dulo m . All ro w and column products in the square Q (2 m ) are semi-circular and equal to the magic constant µ = r 4 ˜ µ , which completes the pro of. Example 4.4. In Figure 6 we show the construction of magic square M D 32 (8) using p ow er squares from Figure 1. W e now show that in our constructions, a giv en semi-magic square ma y admit t wo distinct magic constants, dep ending on the order in which the pro ducts are tak en. 9 Let c ∈ [0 , m − 1] and define the row and column pro ducts by ρ uv i,c = q uv i,i + c q uv i,i + c +1 . . . q uv i,i + c − 1 , σ uv j,c = q uv j − c,j q uv j − c − 1 ,j . . . q uv j − c +1 ,j , where all subscripts are taken mo dulo m . Th us ρ uv i = ρ uv i, 0 and σ uv j = σ uv j, 0 in Q uv . Observ e that in Construction 4.2 (that is, for reflection squares Q uv , i.e. u + v o dd) we hav e ρ uv i,c = ( r 2 ˜ µ − m/ 2 , if c is even , r − 2 ˜ µ + m/ 2 , if c is o dd , σ uv j,c = ( r 2 ˜ µ − m/ 2 , if c is even , r − 2 ˜ µ + m/ 2 , if c is o dd . Consequen tly , in the pro of of Theorem 4.3, the resulting magic constant dep ends on the chosen ordering. Indeed, ρ i = ρ u 1 i, 0 ρ u 2 i,c = ( r 4 ˜ µ , if c is even , r m , if c is o dd , and similarly for the column pro ducts, σ j = σ 1 v j, 0 σ 2 v j,c = ( r 4 ˜ µ , if c is even , r m , if c is o dd . Th us, tw o distinct magic constan ts may arise from the same semi-magic square, dep ending on the order in whic h the pro ducts are taken. F or example, in Figure 6 we hav e ρ u 1 i, 0 ρ u 2 i, 1 = σ 1 v j, 0 σ 1 v j, 1 = r 4 whic h is also a magic constant. 5 Construction for m o dd W e now present a general construction for m o dd. Let E ( m ) = ( e i,j ) m × m , O ( m ) = ( o i,j ) m × m and T ( m ) = ( t i,j ) m × m b e the p o w er squares defined in Section 3. Construction 5.1. Squar e Q 11 Let E ( m ) = ( e i,j ) m × m b e the p ow er square defined in Section 3. Let q 11 i,j = r e i,j , 10 for any i, j ∈ [0 , m − 1]. Therefore ρ 11 i = r e i,i r e i,i +1 . . . r e i,m − 1 r e i, 0 r e i, 1 . . . r e i,i − 1 = r 2 ˜ µ , σ 11 j = r e j,j r e j − 1 ,j . . . r e 0 ,j r e m − 1 ,j r e m − 2 ,j . . . r e j +1 ,j = r 2 ˜ µ . An example is shown in Figure 7. r e 1 , 1 r e 1 , 2 r e 1 , 3 r e 1 , 4 r e 1 , 5 r e 2 , 1 r e 2 , 2 r e 2 , 3 r e 2 , 4 r e 2 , 5 r e 3 , 1 r e 3 , 2 r e 3 , 3 r e 3 , 4 r e 3 , 5 r e 4 , 1 r e 4 , 2 r e 4 , 3 r e 4 , 4 r e 4 , 5 r e 5 , 1 r e 5 , 2 r e 5 , 3 r e 5 , 4 r e 5 , 5 = r 34 r 48 r 2 r 16 r 30 r 46 r 10 r 14 r 28 r 32 r 8 r 12 r 26 r 40 r 44 r 20 r 24 r 38 r 42 r 6 r 22 r 36 r 0 r 4 r 18 Figure 7: Square Q 11 (5) Construction 5.2. Squar e Q 22 Let q 22 i,j =      r e i,j s, for j = i, r t i,j s, for j = i + 1 , r o i,j , otherwise . Observ e that for any i ∈ [0 , m − 1] we hav e e i,i − t i,i +1 + m − 1 X s =2 o i,i + s = 2 ˜ m i,i − ( − 2 ˜ m i,i +1 + m − 2) + m − 1 X s =2 (2 ˜ m i,i + s + 1) = − ( m − 2) + ( m − 2) + m − 1 X s =0 2 ˜ m i,i + s = 2 ˜ µ, where the subscripts are taken modulo m . Moreo ver, for any j ∈ [0 , m − 1] we ha ve e j,j − t j − 1 ,j + m − 1 X s =2 o j − s,j = 2 ˜ m j,j − ( − 2 ˜ m j − 1 ,j + m − 2) + m − 1 X s =2 (2 ˜ m j − s,j + 1) = − ( m − 2) + ( m − 2) + m − 1 X s =0 2 ˜ m j − s,j = 2 ˜ µ, 11 where the subscripts are taken mo dulo m . Therefore, ρ 22 i = r e i,i sr t i,i +1 sr o i,i +2 r o i,i +2 . . . r o i,m − 1 r o i, 0 r o i, 1 . . . r o i,i − 1 = r e i,i r − t i,i +1 r o i,i +2 r o i,i +2 . . . r o i,m − 1 r o i, 0 r o i, 1 . . . r o i,i − 1 = r 2 ˜ µ , σ 22 j = r e j,j sr t j − 1 ,j sr o j − 2 ,j r o j − 3 ,j . . . r o 0 ,j r o m − 1 ,j r o m − 2 ,j . . . r o j +1 ,j = r e j,j r − t j − 1 ,j r o j − 2 ,j r o j − 3 ,j . . . r o 0 ,j r o m − 1 ,j r o m − 2 ,j . . . r o j +1 ,j = r 2 ˜ µ . An example is shown in Figure 8. r e 1 , 1 s r t 1 , 2 s r o 1 , 3 r o 1 , 4 r o 1 , 5 r o 2 , 1 r e 2 , 2 s r t 2 , 3 s r o 2 , 4 r o 2 , 5 r o 3 , 1 r o 3 , 2 r e 3 , 3 s r t 3 , 4 s r o 3 , 5 r o 4 , 1 r o 4 , 2 r o 4 , 3 r e 4 , 4 s r t 4 , 5 s r t 5 , 1 s r o 5 , 2 s r o 5 , 3 r o 5 , 4 r e 5 , 5 s = r 34 s r 5 s r 3 r 17 r 31 r 47 r 10 s r 39 s r 29 r 33 r 9 r 13 r 26 s r 13 s r 45 r 21 r 25 r 39 r 42 s r 47 s r 31 s r 37 r 1 r 5 r 18 s Figure 8: Square Q 22 (5) Construction 5.3. Squar e Q 12 and Q 21 W e construct now the squares Q 12 and Q 21 . Let q 12 i,j =                r o i,j , j = i, r e i,j s, i + j odd , j ∈ [ i + 1 , m − 1] , r t i,j s, i + j ev en , j ∈ [ i + 2 , m − 1] , r e i,j s, i + j ev en , j ∈ [0 , i − 1] , r t i,j s, i + j o dd , j ∈ [0 , i − 1] . An example is shown in Figure 9. r o 1 , 1 r e 1 , 2 s r t 1 , 3 s r e 1 , 4 s r t 1 , 5 s r t 2 , 1 s r o 2 , 2 r e 2 , 3 s r t 2 , 4 s r e 2 , 5 s r e 3 , 1 s r t 3 , 2 s r o 3 , 3 r e 3 , 4 s r t 3 , 5 s r t 4 , 1 s r e 4 , 2 s r t 4 , 3 s r o 4 , 4 r e 4 , 5 s r e 5 , 1 s r t 5 , 2 s r e 5 , 3 s r t 5 , 4 s r o 5 , 5 = r 35 r 48 s r 1 s r 16 s r 23 s r 7 s r 11 r 14 s r 25 s r 32 s r 8 s r 41 s r 27 r 40 s r 9 s r 33 s r 24 s r 15 s r 43 r 6 s r 22 s r 17 s r 0 s r 49 s r 19 Figure 9: Square Q 12 (5) 12 q 21 i,j =                r o i,j +1 , j = i, r e i,j +1 s, i + j odd , j ∈ [ i + 2 , m − 1] , r t i,j +1 s, i + j ev en , j ∈ [ i + 1 , m − 1] , r e i,j +1 s, i + j ev en , j ∈ [0 , i − 1] , r t i,j +1 s, i + j odd , j ∈ [0 , i − 1] . Observ e that for any i, j, l ∈ [0 , m − 1], l  = i and l  = j w e ha ve e i,l − t i,l +1 = 2 ˜ m i,l − ( − 2 ˜ m i,l +1 + m − 2) = 2 ˜ m i,l + 2 ˜ m i,l +1 − m + 2 , e l,j − t l − 1 ,j = 2 ˜ m l,j − ( − 2 ˜ m l − 1 ,j + m − 2) = 2 ˜ m l,j + 2 ˜ m l − 1 ,j − m + 2 . where the subscripts are taken mo dulo m . Therefore, ρ 12 i = r o i,i r e i,i +1 sr t i,i +2 sr e i,i +3 sr t i,i +4 s . . . r e i,i − 2 sr t i,i − 1 s = r 2 ˜ m i,i +1 r e i,i +1 r − t i,i +2 r e i,i +3 r − t i,i +4 . . . r e i,i − 2 r − t i,i − 1 = r 2 ˜ µ − ( m − 2)( m − 1) / 2+1 , σ 12 j = r o j,j r e j − 1 ,j sr t j − 2 ,j sr e j − 3 ,j sr t j − 4 ,j s . . . r e j +2 ,j sr t j +1 ,j s = r 2 ˜ m j,j +1 r e j − 1 ,j r − t j − 2 ,j r e j − 3 ,j r − t j − 4 ,j . . . r e j +2 ,j r − t j +1 ,j = r 2 ˜ µ − ( m − 2)( m − 1) / 2+1 , ρ 21 i = r o i,i +1 r e i,i +2 sr t i,i +3 sr e i,i +4 sr t i,i +5 s . . . r e i,i − 2 sr t i,i − 1 s = r 2 ˜ m i,i +1 +1 r e i,i +2 r − t i,i +3 r e i,i +4 r − t i,i +5 . . . r e i,i − 2 r − t i,i − 1 = r 2 ˜ µ − ( m − 2)( m − 1) / 2+1 , σ 21 j = r o j,j +1 r e j − 1 ,j +1 sr t j − 2 ,j +1 sr e j − 3 ,j +1 sr t j − 4 ,j +1 s . . . r e j − 2 ,j +1 sr t j − 1 ,j +1 s r 2 ˜ m j,j +1 +1 r e j − 1 ,j +1 r − t j − 2 ,j +1 r e j − 3 ,j +1 r − t j − 4 ,j +1 . . . r e j − 2 ,j +1 r − t j − 1 ,j +1 = r 2 ˜ µ − ( m − 2)( m − 1) / 2+1 . An example is shown in Figure 10. r e 1 , 2 s r t 1 , 3 s r e 1 , 4 s r t 1 , 5 s r o 1 , 1 r o 2 , 2 r e 2 , 3 s r t 2 , 4 s r e 2 , 5 s r t 2 , 1 s r t 3 , 2 s r o 3 , 3 r e 3 , 4 s r t 3 , 5 s r e 3 , 1 s r e 4 , 2 s r t 4 , 3 s r o 4 , 4 r e 4 , 5 s r t 4 , 1 s r t 5 , 2 s r e 5 , 3 s r t 5 , 4 s r o 5 , 5 r e 5 , 1 s = r 2 s r 37 s r 30 s r 19 s r 49 r 15 r 28 s r 21 s r 46 s r 43 s r 27 s r 41 r 44 s r 45 s r 12 s r 38 s r 11 s r 7 r 20 s r 29 s r 3 s r 4 s r 35 s r 13 r 36 s Figure 10: Square Q 21 (5) 13 W e are now ready to state our main result of this section. Theorem 5.4. Ther e exists a semi-cir cularly D 2 m 2 -semi-magic squar e Q (2 m ) for every o dd m , m > 1 . Pr o of. Let Q u,v b e the four squares obtained b y Constructions 5.1, 5.2 and 5.3 W e will glue them to obtain a square Q as in Figure 5. Eac h ro w pro duct is now p erformed as ρ i = ( q u 1 i,i q u 1 i,i +1 . . . q u 1 i,m − 1 q u 1 i, 0 q u 1 i, 1 . . . q u 1 i,i − 1 ) ( q u 2 i,i q u 2 i,i +1 . . . q u 2 i,m − 1 q u 2 i, 0 q u 2 i, 1 . . . q u 2 i,i − 1 ) = ρ u 1 i ρ u 2 i = r 2 ˜ µ r 2 ˜ µ − ( m − 2)( m − 1) / 2+1 = r 4 ˜ µ − ( m − 2)( m − 1) / 2+1 and the column pro ducts as σ j = ( q 1 v j,j q 1 v j − 1 ,j . . . q 1 v 0 ,j q 1 v m − 1 ,j q 1 v m − 2 ,j . . . q 1 v j +1 ,j ) ( q 2 v j,j q 2 v j − 1 ,j . . . q 2 v 0 ,j q 2 v m − 1 ,j q 2 v m − 2 ,j . . . q 2 v j +1 ,j ) = σ 1 v i σ 2 v i = r 2 ˜ µ + m/ 2 r 2 ˜ µ − ( m − 2)( m − 1) / 2+1 = r 4 ˜ µ − ( m − 2)( m − 1) / 2+1 , where the subscripts are taken mo dulo m . All ro w and column products in the square Q (2 m ) are semi-circular and equal to the magic constant µ = r 4 ˜ µ − ( m − 2)( m − 1) / 2+1 , whic h completes the pro of. Example 5.5. In Figure 11 w e sho w the construction of magic square M D 50 (10) using p ow er squares from Figure 2. As in Section 4, w e now analyze the case where m is o dd and show that the same situation o ccurs. Let the row pro ducts ρ uv i,c and column σ uv i,c pro ducts b e defined as in Section 4. Observ e that in Construction 5.3 (that is, for u + v o dd) we hav e ρ uv i,c = ( r 2 ˜ µ − ( m − 2)( m − 1) / 2+1 if c is o dd or c = 0 , r − 2 ˜ µ +( m − 2)( m − 1) / 2 − 1 if c is even and c  = 0 , σ uv j,c = ( r 2 ˜ µ − ( m − 2)( m − 1) / 2+1 if c is o dd or c = 0 , r − 2 ˜ µ +( m − 2)( m − 1) / 2 − 1 if c is even and c  = 0 , Consequen tly , in the pro of of Theorem 5.4, the resulting magic constant again dep ends on the c hosen ordering. Indeed, ρ i = ρ u 1 i, 0 ρ u 2 i,c = ( r 4 ˜ µ − ( m − 2)( m − 1) / 2+1 , if c is o dd or c = 0 , r ( m − 2)( m − 1) / 2 − 1 , if c is even and c  = 0 , 14 r 34 r 48 r 2 r 16 r 30 r 35 r 48 s r 1 s r 16 s r 23 s r 46 r 10 r 14 r 28 r 32 r 7 s r 11 r 14 s r 25 s r 32 s r 8 r 12 r 26 r 40 r 44 r 8 s r 41 s r 27 r 40 s r 9 s r 20 r 24 r 38 r 42 r 6 r 33 s r 24 s r 15 s r 43 r 6 s r 22 r 36 r 0 r 4 r 18 r 22 s r 17 s r 0 s r 49 s r 19 r 49 r 2 s r 37 s r 30 s r 19 s r 34 s r 5 s r 3 r 17 r 31 r 43 s r 15 r 28 s r 21 s r 46 s r 47 r 10 s r 39 s r 29 r 33 r 12 s r 27 s r 41 r 44 s r 45 s r 9 r 13 r 26 s r 13 s r 45 r 29 s r 38 s r 11 s r 7 r 20 s r 21 r 25 r 39 r 42 s r 47 s r 36 s r 3 s r 4 s r 35 s r 23 r 31 s r 37 r 1 r 5 r 18 s Figure 11: S M S D 50 (10) with the magic constant µ = r 5 and similarly for the column pro ducts, σ j = σ 1 v j, 0 σ 2 v j,c = ( r 4 ˜ µ − ( m − 2)( m − 1) / 2+1 , if c is o dd or c = 0 , r ( m − 2)( m − 1) / 2 − 1 , if c is even and c  = 0 . Th us, also in the case where m is o dd, tw o distinct magic constants may arise from the same semi-magic square, dep ending on the order in whic h the pro ducts are taken. 6 Conclusion W e summarize our tw o previous results b elow by stating the necessary and sufficien t conditions for the existence of S M S D 2 k ( n ). Theorem 6.1. Ther e exists a Γ -semi-magic squar e S M S Γ ( n ) , wher e Γ is a dihe dr al gr oup, if and only if n is even and n ≥ 4 . Pr o of. It can be verified easily that suc h S M S Γ (2) does not exist. The necessit y follo ws from Theorem 2.5. The existence follows from Theorem 4.3 for n ≡ 0 (mo d 4) and from Theorem 5.4 for n ≡ 2 (mo d 4). Because w e w ere unable to find linearly Γ-semi-magic squares, we p ose an op en problem here. Op en Problem 1. Construct linearly Γ-semi-magic squares S M S Γ ( n ), where Γ is a dihedral group, for ev ery n ≥ 4. Finally , b ecause all our squares are Γ-semi-magic but not Γ-magic, we con- clude with the following. 15 Op en Problem 2. Construct Γ-magic squares M S Γ ( n ), where Γ is a dihedral group, for every even n , n ≥ 4. Example 6.2. In Figure 12 we show a magic rectangle M D 4 (2 , 4) = ( a i,j ) 2 × 4 . Namely , observe that a 1 , 1 a 1 , 2 a 1 , 3 a 1 , 4 = a 2 , 1 a 2 , 2 a 2 , 4 a 2 , 3 = r 2 and a 1 , 1 a 2 , 1 = a 1 , 2 a 2 , 2 = a 1 , 3 a 2 , 3 = a 2 , 4 a 1 , 4 = r 1 . r 0 r 3 r 0 s r 1 s r 1 r 2 r 3 s r 2 s Figure 12: A D 4 -semi-magic rectangle M R D 4 (2 , 4) Motiv ated by Example 6.2, we also state the follo wing op en problem. Op en Problem 3. Characterize Γ-magic rectangles M S Γ ( m, n ), where Γ is a dihedral group of order mn/ 2. In our constructions, we sho wed that it is p ossible to obtain different m agic constan ts for a given magic rectangle S M S D 2 m 2 (2 m ). Some of these constants dep end on ˜ µ , which is the magic constan t of a numerical magic square ˜ M ( m ). Observ e that the square ˜ M ( m ) with the magic constan t ˜ µ can b e transformed in to a magic square c M ( m ) with a differen t magic constant b µ via a uniform translation. Sp ecifically , adding x to each entry of ˜ M ( m ) pro duces c M ( m ) with b µ = ˜ µ + mx. Example 6.3. In Figure 13, the magic square c M (5) has a magic constant b µ = 70. Therefore, we obtain an S M S D 50 (10) with a magic constant µ = r 25 . 16 18 25 2 9 16 24 6 8 15 17 5 7 14 21 23 11 13 20 22 4 12 19 26 3 10 (a) c M (5), b µ = 70 36 0 4 18 32 48 12 16 30 34 10 14 28 42 46 22 26 40 44 8 24 38 2 6 20 (b) E (5) 37 1 5 19 33 49 13 17 31 35 11 15 29 43 47 23 27 41 45 9 25 39 3 7 21 (c) O (5) 17 3 49 35 21 5 41 37 23 19 43 39 25 11 7 31 27 13 9 45 29 15 1 47 33 (d) T (5) r 36 r 0 r 4 r 18 r 32 r 37 r 0 s r 49 s r 18 s r 21 s r 48 r 12 r 16 r 30 r 34 r 5 s r 13 r 16 s r 23 s r 34 s r 10 r 14 r 28 r 42 r 46 r 10 s r 39 s r 29 r 42 s r 7 s r 22 r 26 r 40 r 44 r 8 r 31 s r 26 s r 13 s r 45 r 8 s r 24 r 38 r 2 r 6 r 20 r 24 s r 15 s r 2 s r 47 s r 21 r 1 r 4 s r 35 s r 32 s r 17 s r 36 s r 3 s r 5 r 19 r 33 r 41 s r 17 r 30 s r 19 s r 48 s r 49 r 12 s r 37 s r 31 r 35 r 14 s r 25 s r 43 r 46 s r 43 s r 11 r 15 r 28 s r 11 s r 47 r 27 s r 40 s r 9 s r 9 r 22 s r 23 r 27 r 39 r 44 s r 45 s r 38 s r 1 s r 6 s r 33 s r 25 r 29 s r 39 r 3 r 7 r 20 s (e) S M S D 50 (10) with magic constan t µ = r 25 Figure 13: Example of E (5), O (5), T (5), and S M S D 50 (10) Th us, for a dihedral group Γ of even order n 2 > 4, w e can define the sp e ctrum of magic c onstants by Sp ec Γ ( n ) = { µ ∈ Γ : there exists an S M S Γ ( n ) with magic constant µ } . Consequen tly , we p ose the following op en problem: Op en Problem 4. Characterize Spec Γ ( n ), where Γ is a dihedral group of order n 2 > 4. 17 7 Statemen ts and declarations The work of the first author w as supp orted by program AGH Universit y of Krak ow under gran t no. 16.16.420.054, funded by the Polish Ministry of Science and Higher Education. The work of the second author was partially supp orted b y program ”Excellence initiativ e – researc h univ ersit y” for the A GH Univ ersity . References [1] S. Cichacz, Partition of Ab elian groups into zero-sum sets by complete mappings and its application to the existence of a magic rectangle set, J. A lgebr. Comb. 61(24) (2025) 112815 [2] S. Cichacz, D. F roncek, Magic squares on Ab elian groups, Discr ete Math. 349(7) (2026), 115033. [3] S. Cichacz, T. Hinc, A magic rectangle set on Ab elian groups and its ap- plication, Discr ete Appl. Math. 288 (2021), 201–210. [4] C. J. Colbourn, J. H. Dinitz, eds., Handb o ok of c ombinatorial designs , sec- ond edn., Discrete Ma thema tics and its Applica tions (Boca Ra- ton) , Chapman & Hill/CRC Press, Bo ca Raton, FL 2007. [5] A. B. Ev ans, Magic rectangles and mo dular magic rectangles, J. Stat.Plann. Infer enc e 51 (1996), 171–180. [6] D. F roncek, Semi-magic dihedral squares of side n ≡ 0 (mo d 4), submitted. [7] F. Morini, M.A. Pellegrini, S. Sora, On a conjecture by Sylwia Cichacz and T omasz Hinc, and a related problem, Discr ete Appl. Math. 367 (2025), 53–67. [8] H. Sun, W. Yihui, Note on Magic Squares and Magic Cub es on Ab elian Groups, J. Math. R es. Exp osition 17(2) (1997), 176–178. [9] S. Y u, T. F eng, H. Liu, Existence of magic rectangle sets ov er finite ab elian groups, J. Combin. Des. 33(9) (2025) 329–337. 18