Magic Squares (33b)

Office Applications and Entertainment, Magic Squares
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12.5 Most Perfect Franklin Pan Magic Squares (16 x 16)
(Barink Restrictions)

12.5.1 Introduction

As discussed for the Most Perfect Franklin Pan Magic Squares of the 8^th order in section 8.4.4 the Barink Restrictions can be defined as the additional property that: any 4 consecutive numbers, starting on any odd place in a row or column, sum to one quarter of the Magic Constant.

12.5.2 Analysis

For i = 1, 5, 9 and 13 (i = row or column number) this is already covered by the fact that a 16^th order Most Perfect Franklin Pan Magic Square consists out of 16 Pan Magic Sub Squares of the 4^th order.

The remaining equations, which should be added to the equations shown in section 12.4.2, are:

a( 3)+a( 4)+a( 5)+a( 6) = s1/4
a( 19)+a( 20)+a( 21)+a( 22) = s1/4
a( 35)+a( 36)+a( 37)+a( 38) = s1/4
a( 51)+a( 52)+a( 53)+a( 54) = s1/4
a( 67)+a( 68)+a( 69)+a( 70) = s1/4
a( 83)+a( 84)+a( 85)+a( 86) = s1/4
a( 99)+a(100)+a(101)+a(102) = s1/4
a(115)+a(116)+a(117)+a(118) = s1/4
a(131)+a(132)+a(133)+a(134) = s1/4
a(147)+a(148)+a(149)+a(150) = s1/4
a(163)+a(164)+a(165)+a(166) = s1/4
a(179)+a(180)+a(181)+a(182) = s1/4
a(195)+a(196)+a(197)+a(198) = s1/4
a(211)+a(212)+a(213)+a(214) = s1/4
a(227)+a(228)+a(229)+a(230) = s1/4
a(243)+a(244)+a(245)+a(246) = s1/4

a( 7)+a( 8)+a( 9)+a( 10) = s1/4
a( 23)+a( 24)+a( 25)+a( 26) = s1/4
a( 39)+a( 40)+a( 41)+a( 42) = s1/4
a( 55)+a( 56)+a( 57)+a( 58) = s1/4
a( 71)+a( 72)+a( 73)+a( 74) = s1/4
a( 87)+a( 88)+a( 89)+a( 90) = s1/4
a(103)+a(104)+a(105)+a(106) = s1/4
a(119)+a(120)+a(121)+a(122) = s1/4
a(135)+a(136)+a(137)+a(138) = s1/4
a(151)+a(152)+a(153)+a(154) = s1/4
a(167)+a(168)+a(169)+a(170) = s1/4
a(183)+a(184)+a(185)+a(186) = s1/4
a(199)+a(200)+a(201)+a(202) = s1/4
a(215)+a(216)+a(217)+a(218) = s1/4
a(231)+a(232)+a(233)+a(234) = s1/4
a(247)+a(248)+a(249)+a(250) = s1/4

a( 11)+a( 12)+a( 13)+a( 14) = s1/4
a( 27)+a( 28)+a( 29)+a( 30) = s1/4
a( 43)+a( 44)+a( 45)+a( 46) = s1/4
a( 59)+a( 60)+a( 61)+a( 62) = s1/4
a( 75)+a( 76)+a( 77)+a( 78) = s1/4
a( 91)+a( 92)+a( 93)+a( 94) = s1/4
a(107)+a(108)+a(109)+a(110) = s1/4
a(123)+a(124)+a(125)+a(126) = s1/4
a(139)+a(140)+a(141)+a(142) = s1/4
a(155)+a(156)+a(157)+a(158) = s1/4
a(171)+a(172)+a(173)+a(174) = s1/4
a(187)+a(188)+a(189)+a(190) = s1/4
a(203)+a(204)+a(205)+a(206) = s1/4
a(219)+a(220)+a(221)+a(222) = s1/4
a(235)+a(236)+a(237)+a(238) = s1/4
a(251)+a(252)+a(253)+a(254) = s1/4

a(33)+a(49)+a(65)+a(81) = s1/4
a(34)+a(50)+a(66)+a(82) = s1/4
a(35)+a(51)+a(67)+a(83) = s1/4
a(36)+a(52)+a(68)+a(84) = s1/4
a(37)+a(53)+a(69)+a(85) = s1/4
a(38)+a(54)+a(70)+a(86) = s1/4
a(39)+a(55)+a(71)+a(87) = s1/4
a(40)+a(56)+a(72)+a(88) = s1/4
a(41)+a(57)+a(73)+a(89) = s1/4
a(42)+a(58)+a(74)+a(90) = s1/4
a(43)+a(59)+a(75)+a(91) = s1/4
a(44)+a(60)+a(76)+a(92) = s1/4
a(45)+a(61)+a(77)+a(93) = s1/4
a(46)+a(62)+a(78)+a(94) = s1/4
a(47)+a(63)+a(79)+a(95) = s1/4
a(48)+a(64)+a(80)+a(96) = s1/4

a( 97)+a(113)+a(129)+a(145) = s1/4
a( 98)+a(114)+a(130)+a(146) = s1/4
a( 99)+a(115)+a(131)+a(147) = s1/4
a(100)+a(116)+a(132)+a(148) = s1/4
a(101)+a(117)+a(133)+a(149) = s1/4
a(102)+a(118)+a(134)+a(150) = s1/4
a(103)+a(119)+a(135)+a(151) = s1/4
a(104)+a(120)+a(136)+a(152) = s1/4
a(105)+a(121)+a(137)+a(153) = s1/4
a(106)+a(122)+a(138)+a(154) = s1/4
a(107)+a(123)+a(139)+a(155) = s1/4
a(108)+a(124)+a(140)+a(156) = s1/4
a(109)+a(125)+a(141)+a(157) = s1/4
a(110)+a(126)+a(142)+a(158) = s1/4
a(111)+a(127)+a(143)+a(159) = s1/4
a(112)+a(128)+a(144)+a(160) = s1/4

a(161)+a(177)+a(193)+a(209) = s1/4
a(162)+a(178)+a(194)+a(210) = s1/4
a(163)+a(179)+a(195)+a(211) = s1/4
a(164)+a(180)+a(196)+a(212) = s1/4
a(165)+a(181)+a(197)+a(213) = s1/4
a(166)+a(182)+a(198)+a(214) = s1/4
a(167)+a(183)+a(199)+a(215) = s1/4
a(168)+a(184)+a(200)+a(216) = s1/4
a(169)+a(185)+a(201)+a(217) = s1/4
a(170)+a(186)+a(202)+a(218) = s1/4
a(171)+a(187)+a(203)+a(219) = s1/4
a(172)+a(188)+a(204)+a(220) = s1/4
a(173)+a(189)+a(205)+a(221) = s1/4
a(174)+a(190)+a(206)+a(222) = s1/4
a(175)+a(191)+a(207)+a(223) = s1/4
a(176)+a(192)+a(208)+a(224) = s1/4

The resulting number of equations can be written in matrix representation as:

→ →
A_F * a = s

which can be reduced, by means of row and column manipulations, and results in following set of linear equations:

a(253) =  514 - a(254) - a(255) - a(256)
a(251) =      - a(252) + a(255) + a(256)
a(250) =      - a(252) + a(254) + a(256)
a(249) =  514 + a(252) - a(254) - a(255) - 2*a(256)
a(247) =      - a(248) + a(255) + a(256)
a(246) =      - a(248) + a(254) + a(256)
a(245) =  514 + a(248) - a(254) - a(255) - 2*a(256)
a(243) =      - a(244) + a(255) + a(256)
a(242) =      - a(244) + a(254) + a(256)
a(241) =  514 + a(244) - a(254) - a(255) - 2*a(256)
a(239) =  514 - a(240) - a(255) - a(256)
a(238) =        a(240) - a(254) + a(256)
a(237) =      - a(240) + a(254) + a(255)
a(236) =        a(240) - a(252) + a(256)
a(235) =  514 - a(240) + a(252) - a(255) - 2*a(256)
a(234) =        a(240) + a(252) - a(254)
a(233) =      - a(240) - a(252) + a(254) + a(255) + a(256)
a(232) =        a(240) - a(248) + a(256)
a(231) =  514 - a(240) + a(248) - a(255) - 2*a(256)
a(230) =        a(240) + a(248) - a(254)
a(229) =      - a(240) - a(248) + a(254) + a(255) + a(256)
a(228) =        a(240) - a(244) + a(256)
a(227) =  514 - a(240) + a(244) - a(255) - 2*a(256)
a(226) =        a(240) + a(244) - a(254)
a(225) =      - a(240) - a(244) + a(254) + a(255) + a(256)
a(224) =  257 - a(254)
a(223) = -257 + a(254) + a(255) + a(256)
a(222) =  257 - a(256)
a(221) =  257 - a(255)
a(220) =  257 + a(252) - a(254) - a(256)
a(219) = -257 - a(252) + a(254) + a(255) + 2*a(256)
a(218) =  257 - a(252)
a(217) =  257 + a(252) - a(255) - a(256)
a(216) =  257 + a(248) - a(254) - a(256)
a(215) = -257 - a(248) + a(254) + a(255) + 2*a(256)
a(214) =  257 - a(248)
a(213) =  257 + a(248) - a(255) - a(256)
a(212) =  257 + a(244) - a(254) - a(256)
a(211) = -257 - a(244) + a(254) + a(255) + 2*a(256)
a(210) =  257 - a(244)
a(209) =  257 + a(244) - a(255) - a(256)
a(208) =  257 - a(240) + a(254) - a(256)
a(207) =  257 + a(240) - a(254) - a(255)
a(206) =  257 - a(240)
a(205) = -257 + a(240) + a(255) + a(256)
a(204) =  257 - a(240) - a(252) + a(254)
a(203) =  257 + a(240) + a(252) - a(254) - a(255) - a(256)
a(202) =  257 - a(240) + a(252) - a(256)
a(201) = -257 + a(240) - a(252) + a(255) + 2*a(256)
a(200) =  257 - a(240) - a(248) + a(254)
a(199) =  257 + a(240) + a(248) - a(254) - a(255) - a(256)
a(198) =  257 - a(240) + a(248) - a(256)
a(197) = -257 + a(240) - a(248) + a(255) + 2*a(256)
a(196) =  257 - a(240) - a(244) + a(254)
a(195) =  257 + a(240) + a(244) - a(254) - a(255) - a(256)
a(194) =  257 - a(240) + a(244) - a(256)
a(193) = -257 + a(240) - a(244) + a(255) + 2*a(256)
a(191) =      - a(192) + a(255) + a(256)
a(190) =        a(192) + a(254) - a(256)
a(189) =  514 - a(192) - a(254) - a(255)
a(188) =        a(192) + a(252) - a(256)
a(187) =      - a(192) - a(252) + a(255) + 2*a(256)
a(186) =        a(192) - a(252) + a(254)
a(185) =  514 - a(192) + a(252) - a(254) - a(255) - a(256)
a(184) =        a(192) + a(248) - a(256)
a(183) =      - a(192) - a(248) + a(255) + 2*a(256)
a(182) =        a(192) - a(248) + a(254)
a(181) =  514 - a(192) + a(248) - a(254) - a(255) - a(256)
a(180) =        a(192) + a(244) - a(256)
a(179) =      - a(192) - a(244) + a(255) + 2*a(256)
a(178) =        a(192) - a(244) + a(254)
a(177) =  514 - a(192) + a(244) - a(254) - a(255) - a(256)
a(176) =      - a(192) + a(240) + a(256)
a(175) =  514 + a(192) - a(240) - a(255) - 2*a(256)
a(174) =      - a(192) + a(240) - a(254) + 2*a(256)
a(173) =        a(192) - a(240) + a(254) + a(255) - a(256)
a(172) =      - a(192) + a(240) - a(252) + 2*a(256)
a(171) =  514 + a(192) - a(240) + a(252) - a(255) - 3*a(256)
a(170) =      - a(192) + a(240) + a(252) - a(254) + a(256)
a(169) =        a(192) - a(240) - a(252) + a(254) + a(255)
a(168) =      - a(192) + a(240) - a(248) + 2*a(256)
a(167) =  514 + a(192) - a(240) + a(248) - a(255) - 3*a(256)
a(166) =      - a(192) + a(240) + a(248) - a(254) + a(256)
a(165) =        a(192) - a(240) - a(248) + a(254) + a(255)
a(164) =      - a(192) + a(240) - a(244) + 2*a(256)
a(163) =  514 + a(192) - a(240) + a(244) - a(255) - 3*a(256)
a(162) =      - a(192) + a(240) + a(244) - a(254) + a(256)
a(161) =        a(192) - a(240) - a(244) + a(254) + a(255)
a(160) =  257 - a(192) - a(254) + a(256)
a(159) = -257 + a(192) + a(254) + a(255)
a(158) =  257 - a(192)
a(157) =  257 + a(192) - a(255) - a(256)
a(156) =  257 - a(192) + a(252) - a(254)
a(155) = -257 + a(192) - a(252) + a(254) + a(255) + a(256)
a(154) =  257 - a(192) - a(252) + a(256)
a(153) =  257 + a(192) + a(252) - a(255) - 2*a(256)
a(152) =  257 - a(192) + a(248) - a(254)
a(151) = -257 + a(192) - a(248) + a(254) + a(255) + a(256)
a(150) =  257 - a(192) - a(248) + a(256)
a(149) =  257 + a(192) + a(248) - a(255) - 2*a(256)
a(148) =  257 - a(192) + a(244) - a(254)
a(147) = -257 + a(192) - a(244) + a(254) + a(255) + a(256)
a(146) =  257 - a(192) - a(244) + a(256)
a(145) =  257 + a(192) + a(244) - a(255) - 2*a(256)
a(144) =  257 + a(192) - a(240) + a(254) - 2*a(256)
a(143) =  257 - a(192) + a(240) - a(254) - a(255) + a(256)
a(142) =  257 + a(192) - a(240) - a(256)
a(141) = -257 - a(192) + a(240) + a(255) + 2*a(256)
a(140) =  257 + a(192) - a(240) - a(252) + a(254) - a(256)
a(139) =  257 - a(192) + a(240) + a(252) - a(254) - a(255)
a(138) =  257 + a(192) - a(240) + a(252) - 2*a(256)
a(137) = -257 - a(192) + a(240) - a(252) + a(255) + 3*a(256)
a(136) =  257 + a(192) - a(240) - a(248) + a(254) - a(256)
a(135) =  257 - a(192) + a(240) + a(248) - a(254) - a(255)
a(134) =  257 + a(192) - a(240) + a(248) - 2*a(256)
a(133) = -257 - a(192) + a(240) - a(248) + a(255) + 3*a(256)
a(132) =  257 + a(192) - a(240) - a(244) + a(254) - a(256)
a(131) =  257 - a(192) + a(240) + a(244) - a(254) - a(255)
a(130) =  257 + a(192) - a(240) + a(244) - 2*a(256)
a(129) = -257 - a(192) + a(240) - a(244) + a(255) + 3*a(256)
a(127) =      - a(128) + a(255) + a(256)
a(126) =        a(128) + a(254) - a(256)
a(125) =  514 - a(128) - a(254) - a(255)

a(124) =        a(128) + a(252) - a(256)
a(123) =      - a(128) - a(252) + a(255) + 2*a(256)
a(122) =        a(128) - a(252) + a(254)
a(121) =  514 - a(128) + a(252) - a(254) - a(255) - a(256)
a(120) =        a(128) + a(248) - a(256)
a(119) =      - a(128) - a(248) + a(255) + 2*a(256)
a(118) =        a(128) - a(248) + a(254)
a(117) =  514 - a(128) + a(248) - a(254) - a(255) - a(256)
a(116) =        a(128) + a(244) - a(256)
a(115) =      - a(128) - a(244) + a(255) + 2*a(256)
a(114) =        a(128) - a(244) + a(254)
a(113) =  514 - a(128) + a(244) - a(254) - a(255) - a(256)
a(112) =      - a(128) + a(240) + a(256)
a(111) =  514 + a(128) - a(240) - a(255) - 2*a(256)
a(110) =      - a(128) + a(240) - a(254) + 2*a(256)
a(109) =        a(128) - a(240) + a(254) + a(255) - a(256)
a(108) =      - a(128) + a(240) - a(252) + 2*a(256)
a(107) =  514 + a(128) - a(240) + a(252) - a(255) - 3*a(256)
a(106) =      - a(128) + a(240) + a(252) - a(254) + a(256)
a(105) =        a(128) - a(240) - a(252) + a(254) + a(255)
a(104) =      - a(128) + a(240) - a(248) + 2*a(256)
a(103) =  514 + a(128) - a(240) + a(248) - a(255) - 3*a(256)
a(102) =      - a(128) + a(240) + a(248) - a(254) + a(256)
a(101) =        a(128) - a(240) - a(248) + a(254) + a(255)
a(100) =      - a(128) + a(240) - a(244) + 2*a(256)
a(99) =  514 + a(128) - a(240) + a(244) - a(255) - 3*a(256)
a( 98) =      - a(128) + a(240) + a(244) - a(254) + a(256)
a( 97) =        a(128) - a(240) - a(244) + a(254) + a(255)
a( 96) =  257 - a(128) - a(254) + a(256)
a( 95) = -257 + a(128) + a(254) + a(255)
a( 94) =  257 - a(128)
a( 93) =  257 + a(128) - a(255) - a(256)
a( 92) =  257 - a(128) + a(252) - a(254)
a( 91) = -257 + a(128) - a(252) + a(254) + a(255) + a(256)
a( 90) =  257 - a(128) - a(252) + a(256)
a( 89) =  257 + a(128) + a(252) - a(255) - 2*a(256)
a( 88) =  257 - a(128) + a(248) - a(254)
a( 87) = -257 + a(128) - a(248) + a(254) + a(255) + a(256)
a( 86) =  257 - a(128) - a(248) + a(256)
a( 85) =  257 + a(128) + a(248) - a(255) - 2*a(256)
a( 84) =  257 - a(128) + a(244) - a(254)
a( 83) = -257 + a(128) - a(244) + a(254) + a(255) + a(256)
a( 82) =  257 - a(128) - a(244) + a(256)
a( 81) =  257 + a(128) + a(244) - a(255) - 2*a(256)
a( 80) =  257 + a(128) - a(240) + a(254) - 2*a(256)
a( 79) =  257 - a(128) + a(240) - a(254) - a(255) + a(256)
a( 78) =  257 + a(128) - a(240) - a(256)
a( 77) = -257 - a(128) + a(240) + a(255) + 2*a(256)
a( 76) =  257 + a(128) - a(240) - a(252) + a(254) - a(256)
a( 75) =  257 - a(128) + a(240) + a(252) - a(254) - a(255)
a( 74) =  257 + a(128) - a(240) + a(252) - 2*a(256)
a( 73) = -257 - a(128) + a(240) - a(252) + a(255) + 3*a(256)
a( 72) =  257 + a(128) - a(240) - a(248) + a(254) - a(256)
a( 71) =  257 - a(128) + a(240) + a(248) - a(254) - a(255)
a( 70) =  257 + a(128) - a(240) + a(248) - 2*a(256)
a( 69) = -257 - a(128) + a(240) - a(248) + a(255) + 3*a(256)
a( 68) =  257 + a(128) - a(240) - a(244) + a(254) - a(256)
a( 67) =  257 - a(128) + a(240) + a(244) - a(254) - a(255)
a( 66) =  257 + a(128) - a(240) + a(244) - 2*a(256)
a( 65) = -257 - a(128) + a(240) - a(244) + a(255) + 3*a(256)
a( 63) =      - a(64) + a(255) + a(256)
a( 62) =        a(64) + a(254) - a(256)
a( 61) =  514 - a(64) - a(254) - a(255)
a( 60) =        a(64) + a(252) - a(256)
a( 59) =      - a(64) - a(252) + a(255) + 2*a(256)
a( 58) =        a(64) - a(252) + a(254)
a( 57) =  514 - a(64) + a(252) - a(254) - a(255) - a(256)
a( 56) =        a(64) + a(248) - a(256)
a( 55) =      - a(64) - a(248) + a(255) + 2*a(256)
a( 54) =        a(64) - a(248) + a(254)
a( 53) =  514 - a(64) + a(248) - a(254) - a(255) - a(256)
a( 52) =        a(64) + a(244) - a(256)
a( 51) =      - a(64) - a(244) + a(255) + 2*a(256)
a( 50) =        a(64) - a(244) + a(254)
a( 49) =  514 - a(64) + a(244) - a(254) - a(255) - a(256)
a( 48) =      - a(64) + a(240) + a(256)
a( 47) =  514 + a(64) - a(240) - a(255) - 2*a(256)
a( 46) =      - a(64) + a(240) - a(254) + 2*a(256)
a( 45) =        a(64) - a(240) + a(254) + a(255) - a(256)
a( 44) =      - a(64) + a(240) - a(252) + 2*a(256)
a( 43) =  514 + a(64) - a(240) + a(252) - a(255) - 3*a(256)
a( 42) =      - a(64) + a(240) + a(252) - a(254) + a(256)
a( 41) =        a(64) - a(240) - a(252) + a(254) + a(255)
a( 40) =      - a(64) + a(240) - a(248) + 2*a(256)
a( 39) =  514 + a(64) - a(240) + a(248) - a(255) - 3*a(256)
a( 38) =      - a(64) + a(240) + a(248) - a(254) + a(256)
a( 37) =        a(64) - a(240) - a(248) + a(254) + a(255)
a( 36) =      - a(64) + a(240) - a(244) + 2*a(256)
a( 35) =  514 + a(64) - a(240) + a(244) - a(255) - 3*a(256)
a( 34) =      - a(64) + a(240) + a(244) - a(254) + a(256)
a( 33) =        a(64) - a(240) - a(244) + a(254) + a(255)
a( 32) =  257 - a(64) - a(254) + a(256)
a( 31) = -257 + a(64) + a(254) + a(255)
a( 30) =  257 - a(64)
a( 29) =  257 + a(64) - a(255) - a(256)
a( 28) =  257 - a(64) + a(252) - a(254)
a( 27) = -257 + a(64) - a(252) + a(254) + a(255) + a(256)
a( 26) =  257 - a(64) - a(252) + a(256)
a( 25) =  257 + a(64) + a(252) - a(255) - 2*a(256)
a( 24) =  257 - a(64) + a(248) - a(254)
a( 23) = -257 + a(64) - a(248) + a(254) + a(255) + a(256)
a( 22) =  257 - a(64) - a(248) + a(256)
a( 21) =  257 + a(64) + a(248) - a(255) - 2*a(256)
a( 20) =  257 - a(64) + a(244) - a(254)
a( 19) = -257 + a(64) - a(244) + a(254) + a(255) + a(256)
a( 18) =  257 - a(64) - a(244) + a(256)
a( 17) =  257 + a(64) + a(244) - a(255) - 2*a(256)
a( 16) =  257 + a(64) - a(240) + a(254) - 2*a(256)
a( 15) =  257 - a(64) + a(240) - a(254) - a(255) + a(256)
a( 14) =  257 + a(64) - a(240) - a(256)
a( 13) = -257 - a(64) + a(240) + a(255) + 2*a(256)
a( 12) =  257 + a(64) - a(240) - a(252) + a(254) - a(256)
a( 11) =  257 - a(64) + a(240) + a(252) - a(254) - a(255)
a( 10) =  257 + a(64) - a(240) + a(252) - 2*a(256)
a(  9) = -257 - a(64) + a(240) - a(252) + a(255) + 3*a(256)
a(  8) =  257 + a(64) - a(240) - a(248) + a(254) - a(256)
a(  7) =  257 - a(64) + a(240) + a(248) - a(254) - a(255)
a(  6) =  257 + a(64) - a(240) + a(248) - 2*a(256)
a(  5) = -257 - a(64) + a(240) - a(248) + a(255) + 3*a(256)
a(  4) =  257 + a(64) - a(240) - a(244) + a(254) - a(256)
a(  3) =  257 - a(64) + a(240) + a(244) - a(254) - a(255)
a(  2) =  257 + a(64) - a(240) + a(244) - 2*a(256)
a(  1) = -257 - a(64) + a(240) - a(244) + a(255) + 3*a(256)

The solutions can be obtained by guessing:

a(64), a(128), a(192), a(240), a(244), a(248), a(252), a(254) ... a(256)

and filling out these guesses in the abovementioned equations.

For distinct integers also following inequalities should be applied:

0 < a(i) =< 256       for i = 1, 2 ... 63,  65 ... 127, 129 ... 191, 193 ... 239, 241 ... 243, 245 ... 247
                          i = 249 ... 251, 253
a(i) ≠ a(j)           for i ≠ j

With a(240), a(252), a(254) ... a(256) constant, an optimized guessing routine (MgcSqr16c), produced 384 Most Perfect Franklin Pan Magic Squares under Barink Restrictions within 276 seconds (ref. Attachment 12.5).

12.5.3 Additional Properties

Patterns A and B, which can be translated (with wrap-around) in either direction, both through 8 x 8 sub squares as well as through the 16 x 16 square have been discussed in detail in section 12.3.3.

12.5.4 Spreadsheet Solution

The linear equations, deducted above, can be applied in an Excel spreadsheet (Ref. CnstrSngl16c).

The red figures have to be “guessed” to construct a Franklin Square of the 16^th order (wrong solutions are obvious).

Index

About the Author

a( 3)+a( 4)+a( 5)+a( 6) = s1/4 a( 19)+a( 20)+a( 21)+a( 22) = s1/4 a( 35)+a( 36)+a( 37)+a( 38) = s1/4 a( 51)+a( 52)+a( 53)+a( 54) = s1/4 a( 67)+a( 68)+a( 69)+a( 70) = s1/4 a( 83)+a( 84)+a( 85)+a( 86) = s1/4 a( 99)+a(100)+a(101)+a(102) = s1/4 a(115)+a(116)+a(117)+a(118) = s1/4 a(131)+a(132)+a(133)+a(134) = s1/4 a(147)+a(148)+a(149)+a(150) = s1/4 a(163)+a(164)+a(165)+a(166) = s1/4 a(179)+a(180)+a(181)+a(182) = s1/4 a(195)+a(196)+a(197)+a(198) = s1/4 a(211)+a(212)+a(213)+a(214) = s1/4 a(227)+a(228)+a(229)+a(230) = s1/4 a(243)+a(244)+a(245)+a(246) = s1/4	a( 7)+a( 8)+a( 9)+a( 10) = s1/4 a( 23)+a( 24)+a( 25)+a( 26) = s1/4 a( 39)+a( 40)+a( 41)+a( 42) = s1/4 a( 55)+a( 56)+a( 57)+a( 58) = s1/4 a( 71)+a( 72)+a( 73)+a( 74) = s1/4 a( 87)+a( 88)+a( 89)+a( 90) = s1/4 a(103)+a(104)+a(105)+a(106) = s1/4 a(119)+a(120)+a(121)+a(122) = s1/4 a(135)+a(136)+a(137)+a(138) = s1/4 a(151)+a(152)+a(153)+a(154) = s1/4 a(167)+a(168)+a(169)+a(170) = s1/4 a(183)+a(184)+a(185)+a(186) = s1/4 a(199)+a(200)+a(201)+a(202) = s1/4 a(215)+a(216)+a(217)+a(218) = s1/4 a(231)+a(232)+a(233)+a(234) = s1/4 a(247)+a(248)+a(249)+a(250) = s1/4	a( 11)+a( 12)+a( 13)+a( 14) = s1/4 a( 27)+a( 28)+a( 29)+a( 30) = s1/4 a( 43)+a( 44)+a( 45)+a( 46) = s1/4 a( 59)+a( 60)+a( 61)+a( 62) = s1/4 a( 75)+a( 76)+a( 77)+a( 78) = s1/4 a( 91)+a( 92)+a( 93)+a( 94) = s1/4 a(107)+a(108)+a(109)+a(110) = s1/4 a(123)+a(124)+a(125)+a(126) = s1/4 a(139)+a(140)+a(141)+a(142) = s1/4 a(155)+a(156)+a(157)+a(158) = s1/4 a(171)+a(172)+a(173)+a(174) = s1/4 a(187)+a(188)+a(189)+a(190) = s1/4 a(203)+a(204)+a(205)+a(206) = s1/4 a(219)+a(220)+a(221)+a(222) = s1/4 a(235)+a(236)+a(237)+a(238) = s1/4 a(251)+a(252)+a(253)+a(254) = s1/4
a(33)+a(49)+a(65)+a(81) = s1/4 a(34)+a(50)+a(66)+a(82) = s1/4 a(35)+a(51)+a(67)+a(83) = s1/4 a(36)+a(52)+a(68)+a(84) = s1/4 a(37)+a(53)+a(69)+a(85) = s1/4 a(38)+a(54)+a(70)+a(86) = s1/4 a(39)+a(55)+a(71)+a(87) = s1/4 a(40)+a(56)+a(72)+a(88) = s1/4 a(41)+a(57)+a(73)+a(89) = s1/4 a(42)+a(58)+a(74)+a(90) = s1/4 a(43)+a(59)+a(75)+a(91) = s1/4 a(44)+a(60)+a(76)+a(92) = s1/4 a(45)+a(61)+a(77)+a(93) = s1/4 a(46)+a(62)+a(78)+a(94) = s1/4 a(47)+a(63)+a(79)+a(95) = s1/4 a(48)+a(64)+a(80)+a(96) = s1/4	a( 97)+a(113)+a(129)+a(145) = s1/4 a( 98)+a(114)+a(130)+a(146) = s1/4 a( 99)+a(115)+a(131)+a(147) = s1/4 a(100)+a(116)+a(132)+a(148) = s1/4 a(101)+a(117)+a(133)+a(149) = s1/4 a(102)+a(118)+a(134)+a(150) = s1/4 a(103)+a(119)+a(135)+a(151) = s1/4 a(104)+a(120)+a(136)+a(152) = s1/4 a(105)+a(121)+a(137)+a(153) = s1/4 a(106)+a(122)+a(138)+a(154) = s1/4 a(107)+a(123)+a(139)+a(155) = s1/4 a(108)+a(124)+a(140)+a(156) = s1/4 a(109)+a(125)+a(141)+a(157) = s1/4 a(110)+a(126)+a(142)+a(158) = s1/4 a(111)+a(127)+a(143)+a(159) = s1/4 a(112)+a(128)+a(144)+a(160) = s1/4	a(161)+a(177)+a(193)+a(209) = s1/4 a(162)+a(178)+a(194)+a(210) = s1/4 a(163)+a(179)+a(195)+a(211) = s1/4 a(164)+a(180)+a(196)+a(212) = s1/4 a(165)+a(181)+a(197)+a(213) = s1/4 a(166)+a(182)+a(198)+a(214) = s1/4 a(167)+a(183)+a(199)+a(215) = s1/4 a(168)+a(184)+a(200)+a(216) = s1/4 a(169)+a(185)+a(201)+a(217) = s1/4 a(170)+a(186)+a(202)+a(218) = s1/4 a(171)+a(187)+a(203)+a(219) = s1/4 a(172)+a(188)+a(204)+a(220) = s1/4 a(173)+a(189)+a(205)+a(221) = s1/4 a(174)+a(190)+a(206)+a(222) = s1/4 a(175)+a(191)+a(207)+a(223) = s1/4 a(176)+a(192)+a(208)+a(224) = s1/4