3. 4 完美幻方的变换及构造完美幻方（兼对称）
长方基砖同行上的数位置变动，行与行交换位置都可以得到新的完美幻方。进一步研究表明：变换长方基砖数的分布，可构造完美幻方（兼对称）。以n=3
×k长方基砖为例，给出的长方基砖数的分布:3,4,1 n=3
×k 时，长方基砖组合完美幻方（兼对称）|
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以n=3
×k=3×3=9长方基砖组合为完美幻方（兼对称）|
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当k+1为双偶数n>3
×k=3×11=33时，给出长方基砖通用方程：G(1,1)=1, G(1,2)=5, G(1,3)=8, G(1,4)=11 G(1,4+t1)=10+3
×t1 t1=1,2,3,…..t10 t10=(k-15)/4 ,G(1,J1)=10+3
×t10+2+3×t2 J1=4+t10+t2 t2=1,2,3,….t20 t20=[(k+1)/4]-1 G(1,(k+1)/2)=(3×k-1)/2 G(1,(k+1)/2+t2)=(3×k-1)/2+2+3×t2 G(1,J2)=(3×k-1)/2+3×t20+3×t1 J2=(k+1)/2+t20+t1 G(1,J3)=(3×k-1)/2+3×t20+3×t10+3×t3 J3=J2+t10+t3 t3=1,2,3,4G(2,1)=2, G(2,2)=4, G(2,3)=7, G(2,4)=10, G(2,4+J4)=11+3
×t4, J4=1,2,3,….(k+1)/2-4 G(2,k+1-J5)=3×k+1-G(2,J5) J5=1,2,3,…(k-1)/2 G(3,k+1-J6)=3×k+1-G(1,J6), J6=1,2,3,…k.令: g(1,j)=G(1,J) g(2,j)=G(2,J), g(3,j)=G(3,J) j=J-(k-1)/2 J=1,2,3
…k 当j<0 时， j=j+k 则：g(1,j), g(2,j),g(3,j) 为(k+1)是双偶数长方基砖通用方程。同理亦可作出(k+1)是单偶数长方基砖通用方程（略）及双偶数长方基砖通用方程（略）。3，4，2 n 为素数时，长方基砖组合完美幻方（兼对称） 长方基砖的设计：数的排列顺序为，先奇数，后偶数。 如：n=5 1,3,5,2,4. n=7 1,3,5.7.2,4,6. n=11 1,3,5,7,9,11,2,4,6,8,10. 长方基砖组合为拉丁方的步骤：第二行（或下一行）的填数从上一行2的下边开始。以n=7为例:
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L(I,J) LT(I,J) H(I,J)=n×(L(I,J)-1)+LT(I,J)
3,4,3 n 为双偶数时，长方基砖组合完美幻方（兼对称）
长方基砖设计：如 n=8
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完美幻方（兼对称）
H(I,J)
从以上可得出：用长方基砖组合方法，可统一构造奇数，双偶数完美幻方（兼对称）。