Office Applications and Entertainment, Magic Squares

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12.4   Most Perfect Franklin Pan Magic Squares (16 x 16)

12.4.1 Introduction

Most Perfect Franklin Pan Magic Squares of order 16 are constructed based on following properties:

1. The 4 x 4 subsquares (16 ea) are Pan Magic;
   Consequently the rows, columns and main diagonals sum to the Magic Constant.
2. The main bent diagonals and all the bent diagonals parallel to it sum to the Magic Constant;
3. The main - and broken diagonals sum to the Magic Constant;
4. All 2 × 2 sub squares sum to one quarter the Magic Constant.

Abovementioned properties are comparable with the properties of the 8 x 8 Most Perfect Franklin Pan Magic Squares as described in detail in section 8.5.3.

12.4.2 Analysis

The properties mentioned in section 12.4.1 above result in following set of linear equations:

The 4 x 4 subsquares (16 ea) are Pan Magic (s2 = s1/4):

a1 =-0.5*s2+a36+a51+a52
a2 = 0.5*s2-a36
a3 = 0.5*s2+a36-a50-a51
a4 = 0.5*s2-a36+a50-a52
a17= 0.5*s2-a51
a18= 0.5*s2-a52
a19=-0.5*s2+a50+a51+a52
a20= 0.5*s2-a50
a33=-a36+a50+a51
a34= a36-a50+a52
a35=  s2-a36-a51-a52
a49=  s2-a50-a51-a52

a5 =-0.5*s2+a40+a55+a56
a6 = 0.5*s2-a40
a7 = 0.5*s2+a40-a54-a55
a8 = 0.5*s2-a40+a54-a56
a21= 0.5*s2-a55
a22= 0.5*s2-a56
a23=-0.5*s2+a54+a55+a56
a24= 0.5*s2-a54
a37=-a40+a54+a55
a38= a40-a54+a56
a39=  s2-a40-a55-a56
a53=  s2-a54-a55-a56

a9 =-0.5*s2+a44+a59+a60
a10= 0.5*s2-a44
a11= 0.5*s2+a44-a58-a59
a12= 0.5*s2-a44+a58-a60
a25= 0.5*s2-a59
a26= 0.5*s2-a60
a27=-0.5*s2+a58+a59+a60
a28= 0.5*s2-a58
a41=-a44+a58+a59
a42= a44-a58+a60
a43=  s2-a44-a59-a60
a57=  s2-a58-a59-a60

a13=-0.5*s2+a48+a63+a64
a14= 0.5*s2-a48
a15= 0.5*s2+a48-a62-a63
a16= 0.5*s2-a48+a62-a64
a29= 0.5*s2-a63
a30= 0.5*s2-a64
a31=-0.5*s2+a62+a63+a64
a32= 0.5*s2-a62
a45=-a48+a62+a63
a46= a48-a62+a64
a47=  s2-a48-a63-a64
a61=  s2-a62-a63-a64

a65 =-0.5*s2+a100+a115+a116
a66 = 0.5*s2-a100
a67 = 0.5*s2+a100-a114-a115
a68 = 0.5*s2-a100+a114-a116
a81 = 0.5*s2-a115
a82 = 0.5*s2-a116
a83 =-0.5*s2+a114+a115+a116
a84 = 0.5*s2-a114
a97 =-a100+a114+a115
a98 = a100-a114+a116
a99 =   s2-a100-a115-a116
a113=   s2-a114-a115-a116

a69 =-0.5*s2+a104+a119+a120
a70 = 0.5*s2-a104
a71 = 0.5*s2+a104-a118-a119
a72 = 0.5*s2-a104+a118-a120
a85 = 0.5*s2-a119
a86 = 0.5*s2-a120
a87 =-0.5*s2+a118+a119+a120
a88 = 0.5*s2-a118
a101=-a104+a118+a119
a102= a104-a118+a120
a103=   s2-a104-a119-a120
a117=   s2-a118-a119-a120

a73 =-0.5*s2+a108+a123+a124
a74 = 0.5*s2-a108
a75 = 0.5*s2+a108-a122-a123
a76 = 0.5*s2-a108+a122-a124
a89 = 0.5*s2-a123
a90 = 0.5*s2-a124
a91 =-0.5*s2+a122+a123+a124
a92 = 0.5*s2-a122
a105=-a108+a122+a123
a106= a108-a122+a124
a107=   s2-a108-a123-a124
a121=   s2-a122-a123-a124

a77 =-0.5*s2+a112+a127+a128
a78 = 0.5*s2-a112
a79 = 0.5*s2+a112-a126-a127
a80 = 0.5*s2-a112+a126-a128
a93 = 0.5*s2-a127
a94 = 0.5*s2-a128
a95 =-0.5*s2+a126+a127+a128
a96 = 0.5*s2-a126
a109=-a112+a126+a127
a110= a112-a126+a128
a111=   s2-a112-a127-a128
a125=   s2-a126-a127-a128

a129=-0.5*s2+a164+a179+a180
a130= 0.5*s2-a164
a131= 0.5*s2+a164-a178-a179
a132= 0.5*s2-a164+a178-a180
a145= 0.5*s2-a179
a146= 0.5*s2-a180
a147=-0.5*s2+a178+a179+a180
a148= 0.5*s2-a178
a161=-a164+a178+a179
a162= a164-a178+a180
a163=   s2-a164-a179-a180
a177=   s2-a178-a179-a180

a133=-0.5*s2+a168+a183+a184
a134= 0.5*s2-a168
a135= 0.5*s2+a168-a182-a183
a136= 0.5*s2-a168+a182-a184
a149= 0.5*s2-a183
a150= 0.5*s2-a184
a151=-0.5*s2+a182+a183+a184
a152= 0.5*s2-a182
a165=-a168+a182+a183
a166= a168-a182+a184
a167=   s2-a168-a183-a184
a181=   s2-a182-a183-a184

a137=-0.5*s2+a172+a187+a188
a138= 0.5*s2-a172
a139= 0.5*s2+a172-a186-a187
a140= 0.5*s2-a172+a186-a188
a153= 0.5*s2-a187
a154= 0.5*s2-a188
a155=-0.5*s2+a186+a187+a188
a156= 0.5*s2-a186
a169=-a172+a186+a187
a170= a172-a186+a188
a171=   s2-a172-a187-a188
a185=   s2-a186-a187-a188

a141=-0.5*s2+a176+a191+a192
a142= 0.5*s2-a176
a143= 0.5*s2+a176-a190-a191
a144= 0.5*s2-a176+a190-a192
a157= 0.5*s2-a191
a158= 0.5*s2-a192
a159=-0.5*s2+a190+a191+a192
a160= 0.5*s2-a190
a173=-a176+a190+a191
a174= a176-a190+a192
a175=   s2-a176-a191-a192
a189=   s2-a190-a191-a192

a193=-0.5*s2+a228+a243+a244
a194= 0.5*s2-a228
a195= 0.5*s2+a228-a242-a243
a196= 0.5*s2-a228+a242-a244
a209= 0.5*s2-a243
a210= 0.5*s2-a244
a211=-0.5*s2+a242+a243+a244
a212= 0.5*s2-a242
a225=-a228+a242+a243
a226= a228-a242+a244
a227=   s2-a228-a243-a244
a241=   s2-a242-a243-a244

a197=-0.5*s2+a232+a247+a248
a198= 0.5*s2-a232
a199= 0.5*s2+a232-a246-a247
a200= 0.5*s2-a232+a246-a248
a213= 0.5*s2-a247
a214= 0.5*s2-a248
a215=-0.5*s2+a246+a247+a248
a216= 0.5*s2-a246
a229=-a232+a246+a247
a230= a232-a246+a248
a231=   s2-a232-a247-a248
a245=   s2-a246-a247-a248

a201=-0.5*s2+a236+a251+a252
a202= 0.5*s2-a236
a203= 0.5*s2+a236-a250-a251
a204= 0.5*s2-a236+a250-a252
a217= 0.5*s2-a251
a218= 0.5*s2-a252
a219=-0.5*s2+a250+a251+a252
a220= 0.5*s2-a250
a233=-a236+a250+a251
a234= a236-a250+a252
a235=   s2-a236-a251-a252
a249=   s2-a250-a251-a252

a205=-0.5*s2+a240+a255+a256
a206= 0.5*s2-a240
a207= 0.5*s2+a240-a254-a255
a208= 0.5*s2-a240+a254-a256
a221= 0.5*s2-a255
a222= 0.5*s2-a256
a223=-0.5*s2+a254+a255+a256
a224= 0.5*s2-a254
a237=-a240+a254+a255
a238= a240-a254+a256
a239=   s2-a240-a255-a256
a253=   s2-a254-a255-a256

The main bent diagonals and all the bent diagonals parallel to it sum to the Magic Constant: 

a1 +a18+a35+a52+a69+a86+a103+a120+a136+a151+a166+a181+a196+a211+a226+a241 = s1
a2 +a19+a36+a53+a70+a87+a104+a121+a137+a152+a167+a182+a197+a212+a227+a242 = s1
a3 +a20+a37+a54+a71+a88+a105+a122+a138+a153+a168+a183+a198+a213+a228+a243 = s1
a4 +a21+a38+a55+a72+a89+a106+a123+a139+a154+a169+a184+a199+a214+a229+a244 = s1
a5 +a22+a39+a56+a73+a90+a107+a124+a140+a155+a170+a185+a200+a215+a230+a245 = s1
a6 +a23+a40+a57+a74+a91+a108+a125+a141+a156+a171+a186+a201+a216+a231+a246 = s1
a7 +a24+a41+a58+a75+a92+a109+a126+a142+a157+a172+a187+a202+a217+a232+a247 = s1
a8 +a25+a42+a59+a76+a93+a110+a127+a143+a158+a173+a188+a203+a218+a233+a248 = s1
a9 +a26+a43+a60+a77+a94+a111+a128+a144+a159+a174+a189+a204+a219+a234+a249 = s1
a10+a27+a44+a61+a78+a95+a112+a113+a129+a160+a175+a190+a205+a220+a235+a250 = s1
a11+a28+a45+a62+a79+a96+a97 +a114+a130+a145+a176+a191+a206+a221+a236+a251 = s1
a12+a29+a46+a63+a80+a81+ a98+a115+a131+a146+a161+a192+a207+a222+a237+a252 = s1
a13+a30+a47+a64+a65+a82+ a99+a116+a132+a147+a162+a177+a208+a223+a238+a253 = s1
a14+a31+a48+a49+a66+a83+a100+a117+a133+a148+a163+a178+a193+a224+a239+a254 = s1
a15+a32+a33+a50+a67+a84+a101+a118+a134+a149+a164+a179+a194+a209+a240+a255 = s1
a16+a17+a34+a51+a68+a85+a102+a119+a135+a150+a165+a180+a195+a210+a225+a256 = s1 

a1 +a32+a47+a62+a77+a92+a107+a122+a138+a155+a172+a189+a206+a223+a240+a241 = s1
a2 +a17+a48+a63+a78+a93+a108+a123+a139+a156+a173+a190+a207+a224+a225+a242 = s1
a3 +a18+a33+a64+a79+a94+a109+a124+a140+a157+a174+a191+a208+a209+a226+a243 = s1
a4 +a19+a34+a49+a80+a95+a110+a125+a141+a158+a175+a192+a193+a210+a227+a244 = s1
a5 +a20+a35+a50+a65+a96+a111+a126+a142+a159+a176+a177+a194+a211+a228+a245 = s1
a6 +a21+a36+a51+a66+a81+a112+a127+a143+a160+a161+a178+a195+a212+a229+a246 = s1
a7 +a22+a37+a52+a67+a82+a97 +a128+a144+a145+a162+a179+a196+a213+a230+a247 = s1
a8 +a23+a38+a53+a68+a83+a98 +a113+a129+a146+a163+a180+a197+a214+a231+a248 = s1
a9 +a24+a39+a54+a69+a84+a99 +a114+a130+a147+a164+a181+a198+a215+a232+a249 = s1
a10+a25+a40+a55+a70+a85+a100+a115+a131+a148+a165+a182+a199+a216+a233+a250 = s1
a11+a26+a41+a56+a71+a86+a101+a116+a132+a149+a166+a183+a200+a217+a234+a251 = s1
a12+a27+a42+a57+a72+a87+a102+a117+a133+a150+a167+a184+a201+a218+a235+a252 = s1
a13+a28+a43+a58+a73+a88+a103+a118+a134+a151+a168+a185+a202+a219+a236+a253 = s1
a14+a29+a44+a59+a74+a89+a104+a119+a135+a152+a169+a186+a203+a220+a237+a254 = s1
a15+a30+a45+a60+a75+a90+a105+a120+a136+a153+a170+a187+a204+a221+a238+a255 = s1
a16+a31+a46+a61+a76+a91+a106+a121+a137+a154+a171+a188+a205+a222+a239+a256 = s1 

a1  +a18 +a35 +a52 +a69 +a86 +a103+a120+a16 +a31 +a46 +a61 +a76 +a91 +a106+a121 = s1
a17 +a34 +a51 +a68 +a85 +a102+a119+a136+a32 +a47 +a62 +a77 +a92 +a107+a122+a137 = s1
a33 +a50 +a67 +a84 +a101+a118+a135+a152+a48 +a63 +a78 +a93 +a108+a123+a138+a153 = s1
a49 +a66 +a83 +a100+a117+a134+a151+a168+a64 +a79 +a94 +a109+a124+a139+a154+a169 = s1
a65 +a82 +a99 +a116+a133+a150+a167+a184+a80 +a95 +a110+a125+a140+a155+a170+a185 = s1
a81 +a98 +a115+a132+a149+a166+a183+a200+a96 +a111+a126+a141+a156+a171+a186+a201 = s1
a97 +a114+a131+a148+a165+a182+a199+a216+a112+a127+a142+a157+a172+a187+a202+a217 = s1
a113+a130+a147+a164+a181+a198+a215+a232+a128+a143+a158+a173+a188+a203+a218+a233 = s1
a129+a146+a163+a180+a197+a214+a231+a248+a144+a159+a174+a189+a204+a219+a234+a249 = s1
a145+a162+a179+a196+a213+a230+a247+a8  +a160+a175+a190+a205+a220+a235+a250+a9   = s1
a161+a178+a195+a212+a229+a246+a7  +a24 +a176+a191+a206+a221+a236+a251+a10 +a25  = s1
a177+a194+a211+a228+a245+a6  +a23 +a40 +a192+a207+a222+a237+a252+a11 +a26 +a41  = s1
a193+a210+a227+a244+a5  +a22 +a39 +a56 +a208+a223+a238+a253+a12+a27  +a42 +a57  = s1
a209+a226+a243+a4  +a21 +a38 +a55 +a72 +a224+a239+a254+a13 +a28+a43  +a58 +a73  = s1
a225+a242+a3  +a20 +a37 +a54 +a71 +a88 +a240+a255+a14 +a29 +a44+a59  +a74 +a89  = s1
a241+a2  +a19 +a36 +a53 +a70 +a87 +a104+a256+a15 +a30 +a45 +a60+a75  +a90 +a105 = s1

a152+a167+a182+a197+a212+a227+a242+a1  +a153+a170+a187+a204+a221+a238+a255+a16  = s1
a168+a183+a198+a213+a228+a243+a2  +a17 +a169+a186+a203+a220+a237+a254+a15 +a32  = s1
a184+a199+a214+a229+a244+a3  +a18 +a33 +a185+a202+a219+a236+a253+a14 +a31 +a48  = s1
a200+a215+a230+a245+a4  +a19 +a34 +a49 +a201+a218+a235+a252+a13 +a30 +a47 +a64  = s1
a216+a231+a246+a5  +a20 +a35 +a50 +a65 +a217+a234+a251+a12 +a29 +a46 +a63 +a80  = s1
a232+a247+a6  +a21 +a36 +a51 +a66 +a81 +a233+a250+a11 +a28 +a45 +a62 +a79 +a96  = s1
a248+a7  +a22 +a37 +a52 +a67 +a82 +a97 +a249+a10 +a27 +a44 +a61 +a78 +a95 +a112 = s1
a8  +a23 +a38 +a53 +a68 +a83 +a98 +a113+a9  +a26 +a43 +a60 +a77 +a94 +a111+a128 = s1
a24 +a39 +a54 +a69 +a84 +a99 +a114+a129+a25 +a42 +a59 +a76 +a93 +a110+a127+a144 = s1
a40 +a55 +a70 +a85 +a100+a115+a130+a145+a41 +a58 +a75 +a92 +a109+a126+a143+a160 = s1
a56 +a71 +a86 +a101+a116+a131+a146+a161+a57 +a74 +a91 +a108+a125+a142+a159+a176 = s1
a72 +a87 +a102+a117+a132+a147+a162+a177+a73 +a90 +a107+a124+a141+a158+a175+a192 = s1
a88 +a103+a118+a133+a148+a163+a178+a193+a89 +a106+a123+a140+a157+a174+a191+a208 = s1
a104+a119+a134+a149+a164+a179+a194+a209+a105+a122+a139+a156+a173+a190+a207+a224 = s1
a120+a135+a150+a165+a180+a195+a210+a225+a121+a138+a155+a172+a189+a206+a223+a240 = s1
a136+a151+a166+a181+a196+a211+a226+a241+a137+a154+a171+a188+a205+a222+a239+a256 = s1

The main - and broken diagonals sum to the Magic Constant; 

a1 +a18+a35+a52+a69+a86+a103+a120+a137+a154+a171+a188+a205+a222+a239+a256 = s1
a2 +a19+a36+a53+a70+a87+a104+a121+a138+a155+a172+a189+a206+a223+a240+a241 = s1
a3 +a20+a37+a54+a71+a88+a105+a122+a139+a156+a173+a190+a207+a224+a225+a242 = s1
a4 +a21+a38+a55+a72+a89+a106+a123+a140+a157+a174+a191+a208+a209+a226+a243 = s1
a5 +a22+a39+a56+a73+a90+a107+a124+a141+a158+a175+a192+a193+a210+a227+a244 = s1
a6 +a23+a40+a57+a74+a91+a108+a125+a142+a159+a176+a177+a194+a211+a228+a245 = s1
a7 +a24+a41+a58+a75+a92+a109+a126+a143+a160+a161+a178+a195+a212+a229+a246 = s1
a8 +a25+a42+a59+a76+a93+a110+a127+a144+a145+a162+a179+a196+a213+a230+a247 = s1
a9 +a26+a43+a60+a77+a94+a111+a128+a129+a146+a163+a180+a197+a214+a231+a248 = s1
a10+a27+a44+a61+a78+a95+a112+a113+a130+a147+a164+a181+a198+a215+a232+a249 = s1
a11+a28+a45+a62+a79+a96+a97 +a114+a131+a148+a165+a182+a199+a216+a233+a250 = s1
a12+a29+a46+a63+a80+a81+a98 +a115+a132+a149+a166+a183+a200+a217+a234+a251 = s1
a13+a30+a47+a64+a65+a82+a99 +a116+a133+a150+a167+a184+a201+a218+a235+a252 = s1
a14+a31+a48+a49+a66+a83+a100+a117+a134+a151+a168+a185+a202+a219+a236+a253 = s1
a15+a32+a33+a50+a67+a84+a101+a118+a135+a152+a169+a186+a203+a220+a237+a254 = s1
a16+a17+a34+a51+a68+a85+a102+a119+a136+a153+a170+a187+a204+a221+a238+a255 = s1 

a1 +a32+a47+a62+a77+a92+a107+a122+a137+a152+a167+a182+a197+a212+a227+a242 = s1
a2 +a17+a48+a63+a78+a93+a108+a123+a138+a153+a168+a183+a198+a213+a228+a243 = s1
a3 +a18+a33+a64+a79+a94+a109+a124+a139+a154+a169+a184+a199+a214+a229+a244 = s1
a4 +a19+a34+a49+a80+a95+a110+a125+a140+a155+a170+a185+a200+a215+a230+a245 = s1
a5 +a20+a35+a50+a65+a96+a111+a126+a141+a156+a171+a186+a201+a216+a231+a246 = s1
a6 +a21+a36+a51+a66+a81+a112+a127+a142+a157+a172+a187+a202+a217+a232+a247 = s1
a7 +a22+a37+a52+a67+a82+a97 +a128+a143+a158+a173+a188+a203+a218+a233+a248 = s1
a8 +a23+a38+a53+a68+a83+a98 +a113+a144+a159+a174+a189+a204+a219+a234+a249 = s1
a9 +a24+a39+a54+a69+a84+a99 +a114+a129+a160+a175+a190+a205+a220+a235+a250 = s1
a10+a25+a40+a55+a70+a85+a100+a115+a130+a145+a176+a191+a206+a221+a236+a251 = s1
a11+a26+a41+a56+a71+a86+a101+a116+a131+a146+a161+a192+a207+a222+a237+a252 = s1
a12+a27+a42+a57+a72+a87+a102+a117+a132+a147+a162+a177+a208+a223+a238+a253 = s1
a13+a28+a43+a58+a73+a88+a103+a118+a133+a148+a163+a178+a193+a224+a239+a254 = s1
a14+a29+a44+a59+a74+a89+a104+a119+a134+a149+a164+a179+a194+a209+a240+a255 = s1
a15+a30+a45+a60+a75+a90+a105+a120+a135+a150+a165+a180+a195+a210+a225+a256 = s1
a16+a31+a46+a61+a76+a91+a106+a121+a136+a151+a166+a181+a196+a211+a226+a241 = s1

Every 2 × 2 sub square sums to one quarter of the Magic Constant:

a(i) + a(i+1) + a(i+16) + a(i+17) = s1/4 with 1 =< i < 240 and i ≠ 16*n for n = 1, 2 ... 15

a(i) + a(i+1) + a(i+16) + a(i-15) = s1/4 with i = 16*n for n = 1, 2 ... 15

a(i) + a(i+1) + a(i+240) + a(i+241) = s1/4 with i = 1, 2 ... 15

a(1) + a(16)   + a(241)   + a(256)   = s1/4

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

          →   →
     AF * 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(250) =      - a(252) + a(254) + a(256)
a(249) =  514 - a(251) - a(254) - a(256)
a(246) =      - a(248) + a(254) + a(256)
a(245) =  514 - a(247) - a(254) - a(256)
a(242) =      - a(244) + a(254) + a(256)
a(241) =  514 - a(243) - a(254) - 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(251) - a(256)
a(234) =        a(240) + a(252) - a(254)
a(233) =      - a(240) + a(251) + a(254)
a(232) =        a(240) - a(248) + a(256)
a(231) =  514 - a(240) - a(247) - a(256)
a(230) =        a(240) + a(248) - a(254)
a(229) =      - a(240) + a(247) + a(254)
a(228) =        a(240) - a(244) + a(256)
a(227) =  514 - a(240) - a(243) - a(256)
a(226) =        a(240) + a(244) - a(254)
a(225) =      - a(240) + a(243) + a(254)
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(251) + a(254) + a(256)
a(218) =  257 - a(252)
a(217) =  257 - a(251)
a(216) =  257 + a(248) - a(254) - a(256)
a(215) = -257 + a(247) + a(254) + a(256)
a(214) =  257 - a(248)
a(213) =  257 - a(247)
a(212) =  257 + a(244) - a(254) - a(256)
a(211) = -257 + a(243) + a(254) + a(256)
a(210) =  257 - a(244)
a(209) =  257 - a(243)
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(251) - a(254)
a(202) =  257 - a(240) + a(252) - a(256)
a(201) = -257 + a(240) + a(251) + a(256)
a(200) =  257 - a(240) - a(248) + a(254)
a(199) =  257 + a(240) - a(247) - a(254)
a(198) =  257 - a(240) + a(248) - a(256)
a(197) = -257 + a(240) + a(247) + a(256)
a(196) =  257 - a(240) - a(244) + a(254)
a(195) =  257 + a(240) - a(243) - a(254)
a(194) =  257 - a(240) + a(244) - a(256)
a(193) = -257 + a(240) + a(243) + 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(251) + a(256)
a(186) =        a(192) - a(252) + a(254)
a(185) =  514 - a(192) - a(251) - a(254)
a(184) =        a(192) + a(248) - a(256)
a(183) =      - a(192) + a(247) + a(256)
a(182) =        a(192) - a(248) + a(254)
a(181) =  514 - a(192) - a(247) - a(254)
a(180) =        a(192) + a(244) - a(256)
a(179) =      - a(192) + a(243) + a(256)
a(178) =        a(192) - a(244) + a(254)
a(177) =  514 - a(192) - a(243) - a(254)
a(175) =  514 - a(176) - a(255) - a(256)
a(174) =        a(176) - a(254) + a(256)
a(173) =      - a(176) + a(254) + a(255)
a(172) =        a(176) - a(252) + a(256)
a(171) =  514 - a(176) - a(251) - a(256)
a(170) =        a(176) + a(252) - a(254)
a(169) =      - a(176) + a(251) + a(254)
a(168) =        a(176) - a(248) + a(256)
a(167) =  514 - a(176) - a(247) - a(256)
a(166) =        a(176) + a(248) - a(254)
a(165) =      - a(176) + a(247) + a(254)
a(164) =        a(176) - a(244) + a(256)
a(163) =  514 - a(176) - a(243) - a(256)
a(162) =        a(176) + a(244) - a(254)
a(161) =      - a(176) + a(243) + a(254)
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(251) + a(254)
a(154) =  257 - a(192) - a(252) + a(256)
a(153) =  257 + a(192) - a(251) - a(256)
a(152) =  257 - a(192) + a(248) - a(254)
a(151) = -257 + a(192) + a(247) + a(254)
a(150) =  257 - a(192) - a(248) + a(256)
a(149) =  257 + a(192) - a(247) - a(256)
a(148) =  257 - a(192) + a(244) - a(254)
a(147) = -257 + a(192) + a(243) + a(254)
a(146) =  257 - a(192) - a(244) + a(256)
a(145) =  257 + a(192) - a(243) - a(256)
a(144) =  257 - a(176) + a(254) - a(256)
a(143) =  257 + a(176) - a(254) - a(255)
a(142) =  257 - a(176)
a(141) = -257 + a(176) + a(255) + a(256)
a(140) =  257 - a(176) - a(252) + a(254)
a(139) =  257 + a(176) - a(251) - a(254)
a(138) =  257 - a(176) + a(252) - a(256)
a(137) = -257 + a(176) + a(251) + a(256)
a(136) =  257 - a(176) - a(248) + a(254)
a(135) =  257 + a(176) - a(247) - a(254)
a(134) =  257 - a(176) + a(248) - a(256)
a(133) = -257 + a(176) + a(247) + a(256)
a(132) =  257 - a(176) - a(244) + a(254)
a(131) =  257 + a(176) - a(243) - a(254)
a(130) =  257 - a(176) + a(244) - a(256)
a(129) = -257 + a(176) + a(243) + 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(251) + a(256)
a(122) =        a(128) - a(252) + a(254)
a(121) =  514 - a(128) - a(251) - a(254)
a(120) =        a(128) + a(248) - a(256)
a(119) =      - a(128) + a(247) + a(256)
a(118) =        a(128) - a(248) + a(254)
a(117) =  514 - a(128) - a(247) - a(254)
a(116) =        a(128) + a(244) - a(256)
a(115) =      - a(128) + a(243) + a(256)
a(114) =        a(128) - a(244) + a(254)
a(113) =  514 - a(128) - a(243) - a(254)
a(111) =  514 - a(112) - a(255) - a(256)
a(110) =        a(112) - a(254) + a(256)
a(109) =      - a(112) + a(254) + a(255)
a(108) =        a(112) - a(252) + a(256)
a(107) =  514 - a(112) - a(251) - a(256)
a(106) =        a(112) + a(252) - a(254)
a(105) =      - a(112) + a(251) + a(254)
a(104) =        a(112) - a(248) + a(256)
a(103) =  514 - a(112) - a(247) - a(256)
a(102) =        a(112) + a(248) - a(254)
a(101) =      - a(112) + a(247) + a(254)
a(100) =        a(112) - a(244) + a(256)
a( 99) =  514 - a(112) - a(243) - a(256)
a( 98) =        a(112) + a(244) - a(254)
a( 97) =      - a(112) + a(243) + a(254)
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(251) + a(254)
a( 90) =  257 - a(128) - a(252) + a(256)
a( 89) =  257 + a(128) - a(251) - a(256)
a( 88) =  257 - a(128) + a(248) - a(254)
a( 87) = -257 + a(128) + a(247) + a(254)
a( 86) =  257 - a(128) - a(248) + a(256)
a( 85) =  257 + a(128) - a(247) - a(256)
a( 84) =  257 - a(128) + a(244) - a(254)
a( 83) = -257 + a(128) + a(243) + a(254)
a( 82) =  257 - a(128) - a(244) + a(256)
a( 81) =  257 + a(128) - a(243) - a(256)
a( 80) =  257 - a(112) + a(254) - a(256)
a( 79) =  257 + a(112) - a(254) - a(255)
a( 78) =  257 - a(112)
a( 77) = -257 + a(112) + a(255) + a(256)
a( 76) =  257 - a(112) - a(252) + a(254)
a( 75) =  257 + a(112) - a(251) - a(254)
a( 74) =  257 - a(112) + a(252) - a(256)
a( 73) = -257 + a(112) + a(251) + a(256)
a( 72) =  257 - a(112) - a(248) + a(254)
a( 71) =  257 + a(112) - a(247) - a(254)
a( 70) =  257 - a(112) + a(248) - a(256)
a( 69) = -257 + a(112) + a(247) + a(256)
a( 68) =  257 - a(112) - a(244) + a(254)
a( 67) =  257 + a(112) - a(243) - a(254)
a( 66) =  257 - a(112) + a(244) - a(256)
a( 65) = -257 + a(112) + a(243) + 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(251) + a(256)
a( 58) =        a(64) - a(252) + a(254)
a( 57) =  514 - a(64) - a(251) - a(254)
a( 56) =        a(64) + a(248) - a(256)
a( 55) =      - a(64) + a(247) + a(256)
a( 54) =        a(64) - a(248) + a(254)
a( 53) =  514 - a(64) - a(247) - a(254)
a( 52) =        a(64) + a(244) - a(256)
a( 51) =      - a(64) + a(243) + a(256)
a( 50) =        a(64) - a(244) + a(254)
a( 49) =  514 - a(64) - a(243) - a(254)
a( 47) =  514 - a(48) - a(255) - a(256)
a( 46) =        a(48) - a(254) + a(256)
a( 45) =      - a(48) + a(254) + a(255)
a( 44) =        a(48) - a(252) + a(256)
a( 43) =  514 - a(48) - a(251) - a(256)
a( 42) =        a(48) + a(252) - a(254)
a( 41) =      - a(48) + a(251) + a(254)
a( 40) =        a(48) - a(248) + a(256)
a( 39) =  514 - a(48) - a(247) - a(256)
a( 38) =        a(48) + a(248) - a(254)
a( 37) =      - a(48) + a(247) + a(254)
a( 36) =        a(48) - a(244) + a(256)
a( 35) =  514 - a(48) - a(243) - a(256)
a( 34) =        a(48) + a(244) - a(254)
a( 33) =      - a(48) + a(243) + a(254)
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(251) + a(254)
a( 26) =  257 - a(64) - a(252) + a(256)
a( 25) =  257 + a(64) - a(251) - a(256)
a( 24) =  257 - a(64) + a(248) - a(254)
a( 23) = -257 + a(64) + a(247) + a(254)
a( 22) =  257 - a(64) - a(248) + a(256)
a( 21) =  257 + a(64) - a(247) - a(256)
a( 20) =  257 - a(64) + a(244) - a(254)
a( 19) = -257 + a(64) + a(243) + a(254)
a( 18) =  257 - a(64) - a(244) + a(256)
a( 17) =  257 + a(64) - a(243) - a(256)
a( 16) =  257 - a(48) + a(254) - a(256)
a( 15) =  257 + a(48) - a(254) - a(255)
a( 14) =  257 - a(48)
a( 13) = -257 + a(48) + a(255) + a(256)
a( 12) =  257 - a(48) - a(252) + a(254)
a( 11) =  257 + a(48) - a(251) - a(254)
a( 10) =  257 - a(48) + a(252) - a(256)
a(  9) = -257 + a(48) + a(251) + a(256)
a(  8) =  257 - a(48) - a(248) + a(254)
a(  7) =  257 + a(48) - a(247) - a(254)
a(  6) =  257 - a(48) + a(248) - a(256)
a(  5) = -257 + a(48) + a(247) + a(256)
a(  4) =  257 - a(48) - a(244) + a(254)
a(  3) =  257 + a(48) - a(243) - a(254)
a(  2) =  257 - a(48) + a(244) - a(256)
a(  1) = -257 + a(48) + a(243) + a(256)

The solutions can be obtained by guessing:

   a( 48), a(64), a(112), a(128), a(176), a(192), a(240), a(243), a(244), a(247), a(248), a(251), a(252) and
   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 ... 47, 49 ... 63, 65 ... 111, 113 ... 127, 129 ...175, 177 ... 191
                          i = 193 ... 239, 241, 242, 245, 246, 249, 250, 253

a(i) ≠ a(j)           for i ≠ j

With a(176), a(192), a(240), a(243), a(244), a(247), a(248), a(251), a(252), a(254) ... a(256) constant, an optimized guessing routine (MgcSqr16b), produced 384 Most Perfect Franklin Pan Magic Squares within 98 seconds (ref. Attachment 12.4).

With also a(176) variable, the same routine produces 3840 Most Perfect Franklin Pan Magic Squares within 980 seconds.

12.4.3 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.

12.4.4 Spreadsheet Solution

The linear equations, deducted above, can be applied in an Excel spreadsheet (Ref. CnstrSngl16b).

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

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