Office Applications and Entertainment, Magic Cubes

IV   Associated Magic Rectangles

IV-1 Order 3 x 5

Based on the linear equations describing an Associated Magic Rectangle of order 3 x 5:

a(11) = 5 * c1 - a(12) - a(13) - a(14) - a(15)
a(10) =     c1 + a(11) - a(15)
a( 9) =     c1 + a(12) - a(14)
a( 8) =     c1
a( 7) = 2 * c1 - a( 9)
a( 6) = 2 * c1 - a(10)
a( 5) = 2 * c1 - a(11)
a( 4) = 2 * c1 - a(12)
a( 3) = 2 * c1 - a(13)
a( 2) = 2 * c1 - a(14)
a( 1) = 2 * c1 - a(15)

a routine can be written to generate Associated Magic Rectangles of order 3 x 5 (ref. MgcSqr35).

Attachment 6.3.9 shows the 16 possible solutions.

IV-2 Order 3 x 7

Based on the linear equations describing an Associated Magic Rectangle of order 3 x 7:

a(15) = 7 * c1 - a(16) - a(17) - a(18) - a(19) - a(20) - a(21)
a(14) =     c1 + a(15) - a(21)
a(13) =     c1 + a(16) - a(20)
a(12) =     c1 + a(17) - a(19)
a(11) =     c1
a(10) = 2 * c1 - a(12)
a( 9) = 2 * c1 - a(13)
a( 8) = 2 * c1 - a(14)
a( 7) = 2 * c1 - a(15)
a( 6) = 2 * c1 - a(16)
a( 5) = 2 * c1 - a(17)
a( 4) = 2 * c1 - a(18)
a( 3) = 2 * c1 - a(19)
a( 2) = 2 * c1 - a(20)
a( 1) = 2 * c1 - a(21)

a routine can be written to generate Associated Magic Rectangles of order 3 x 7 (ref. MgcSqr37).

Attachment 6.3.15 shows the 18 solutions for a(21) = 12 and a(20) = 1.

IV-3 Order 5 x 7

Based on the linear equations describing an Associated Magic Rectangle of order 5 x 7:

a(22) = 7 * c1 - a(23) - a(24) - a(25) - a(26) - a(27) - a(28)
a(21) =     c1 + a(22) - a(28) + a(29) - a(35)
a(20) =     c1 + a(23) - a(27) + a(30) - a(34)
a(19) =     c1 + a(24) - a(26) + a(31) - a(33)
a(18) =     c1
a(17) = 2 * c1 - a(19)
a(16) = 2 * c1 - a(20)
a(15) = 2 * c1 - a(21)
a(14) = 2 * c1 - a(22)
a(13) = 2 * c1 - a(23)
a(12) = 2 * c1 - a(24)
a(11) = 2 * c1 - a(25)
a(10) = 2 * c1 - a(26)
a( 9) = 2 * c1 - a(27)
a( 8) = 2 * c1 - a(28)
a( 7) = 2 * c1 - a(29)
a( 6) = 2 * c1 - a(30)
a( 5) = 2 * c1 - a(31)
a( 4) = 2 * c1 - a(32)
a( 3) = 2 * c1 - a(33)
a( 2) = 2 * c1 - a(34)
a( 1) = 2 * c1 - a(35)

a routine can be written to generate Associated Magic Rectangles of order 5 x 7 (ref. MgcSqr57).

Attachment 6.3.17 shows 16 of the possible solutions (> 38142).

IV-4 Order 3 x 5 x 7

Based on the linear equations describing an Associated Magic Rectangle of order 3 x 5 x 7:

a(99) = 7 * c1 - a(100) - a(101) - a(102) - a(103) - a(104) - a(105)
a(92) = 7 * c1 - a(93) - a(94) - a(95) - a( 96) - a(97) - a(98)
a(85) = 7 * c1 - a(86) - a(87) - a(88) - a( 89) - a(90) - a(91)
a(78) = 7 * c1 - a(79) - a(80) - a(81) - a( 82) - a(83) - a(84)
a(77) = 5 * c1 - a(84) - a(91) - a(98) - a(105)
a(76) = 5 * c1 - a(83) - a(90) - a(97) - a(104)
a(75) = 5 * c1 - a(82) - a(89) - a(96) - a(103)
a(74) = 5 * c1 - a(81) - a(88) - a(95) - a(102)
a(73) = 5 * c1 - a(80) - a(87) - a(94) - a(101)
a(72) = 5 * c1 - a(79) - a(86) - a(93) - a(100)
a(71) = 5 * c1 - a(78) - a(85) - a(92) - a( 99)
a(70) = 6 * c1 - a(78) - a(85) - a(92) - a( 99) - a(105)
a(69) = 6 * c1 - a(79) - a(86) - a(93) - a(100) - a(104)
a(68) = 6 * c1 - a(80) - a(87) - a(94) - a(101) - a(103)
a(67) = 6 * c1 - a(81) - a(88) - a(95) - 2 * a(102)
a(66) = 6 * c1 - a(82) - a(89) - a(96) - a(101) - a(103)
a(65) = 6 * c1 - a(83) - a(90) - a(97) - a(100) - a(104)
a(64) = 6 * c1 - a(84) - a(91) - a(98) - a( 99) - a(105)
a(63) =     c1 + a(78) - a(98)
a(62) =     c1 + a(79) - a(97)
a(61) =     c1 + a(80) - a(96)
a(60) =     c1 + a(81) - a(95)
a(59) =     c1 + a(82) - a(94)
a(58) =     c1 + a(83) - a(93)
a(57) =     c1 + a(84) - a(92)
a(56) =     c1 + a(85) - a(91)
a(55) =     c1 + a(86) - a(90)
a(54) =     c1 + a(87) - a(89)
a(53) =     c1

a52 = 2 * c1 - a54 a51 = 2 * c1 - a55 a50 = 2 * c1 - a56 a49 = 2 * c1 - a57 a48 = 2 * c1 - a58 a47 = 2 * c1 - a59 a46 = 2 * c1 - a60 a45 = 2 * c1 - a61 a44 = 2 * c1 - a62 a43 = 2 * c1 - a63 a42 = 2 * c1 - a64 a41 = 2 * c1 - a65 a40 = 2 * c1 - a66

a39 = 2 * c1 - a67 a38 = 2 * c1 - a68 a37 = 2 * c1 - a69 a36 = 2 * c1 - a70 a35 = 2 * c1 - a71 a34 = 2 * c1 - a72 a33 = 2 * c1 - a73 a32 = 2 * c1 - a74 a31 = 2 * c1 - a75 a30 = 2 * c1 - a76 a29 = 2 * c1 - a77 a28 = 2 * c1 - a78 a27 = 2 * c1 - a79

a26 = 2 * c1 - a80 a25 = 2 * c1 - a81 a24 = 2 * c1 - a82 a23 = 2 * c1 - a83 a22 = 2 * c1 - a84 a21 = 2 * c1 - a85 a20 = 2 * c1 - a86 a19 = 2 * c1 - a87 a18 = 2 * c1 - a88 a17 = 2 * c1 - a89 a16 = 2 * c1 - a90 a15 = 2 * c1 - a91 a14 = 2 * c1 - a92

a13 = 2 * c1 - a93 a12 = 2 * c1 - a94 a11 = 2 * c1 - a95 a10 = 2 * c1 - a96 a 9 = 2 * c1 - a97 a 8 = 2 * c1 - a98 a 7 = 2 * c1 - a99 a 6 = 2 * c1 - a100 a 5 = 2 * c1 - a101 a 4 = 2 * c1 - a102 a 3 = 2 * c1 - a103 a 2 = 2 * c1 - a104 a 1 = 2 * c1 - a105

a routine can be written to generate Associated Magic Rectangles of order 3 x 5 x 7.

However the total number of independent variables (24) is to large to obtain results within a reasonable time.

Attachment 6.3.19 shows, based on rotation and reflection, one class (8 ea) of possible solutions.