In the traditional 3 x 3 Magic Square the digital sum ("magic" constant) is 15. The square may be duplicated such that each cell has two identical digits forming a
T U combination (eg 11,22) with the common value being 165 .
The digits 1 to 9 may be placed in each cell so that each Tens digit and each Units digit is represented once in the 3 x 3 grid with no digit reappearing in the same column or row; the column and row totals are to have the same value.
While I can offer a solution with all sums but one major diagonal having the same "magic" constant, can you do similar, or even better, offer 8 equal digital sums?
Dim tens(3, 3), units(3, 3), goal
Dim usedT(9), usedU(9)
Private Sub Form_Load()
Me.Visible = True
place 1, 1
End Sub
Sub place(row, col)
For t = 1 To 9
DoEvents
If usedT(t) = 0 Then
goodT = 1
For r = 1 To row - 1
If tens(r, col) = t Then goodT = 0: Exit For
If units(r, col) = t Then goodT = 0: Exit For
Next r
For c = 1 To col - 1
If tens(row, c) = t Then goodT = 0: Exit For
If units(row, c) = t Then goodT = 0: Exit For
Next c
tens(row, col) = t
If col = 3 And row = 1 Then
If tens(1, 3) < tens(1, 1) Then goodT = 0
End If
If col = 1 And row = 3 Then
If tens(3, 1) < tens(1, 1) Or tens(3, 1) < tens(1, 3) Then goodT = 0
End If
If col = 3 And row = 3 Then
If tens(3, 3) < tens(1, 1) Then goodT = 0
End If
If goodT Then
usedT(t) = 1
For u = 1 To 9
If usedU(u) = 0 Then
goodU = 1
For r = 1 To row - 1
If tens(r, col) = u Then goodU = 0: Exit For
If units(r, col) = u Then goodU = 0: Exit For
Next r
For c = 1 To col - 1
If tens(row, c) = u Then goodU = 0: Exit For
If units(row, c) = u Then goodU = 0: Exit For
Next c
If tens(row, col) = u Then goodU = 0
If goodU Then
usedU(u) = 1
units(row, col) = u
If col = 3 Then
Select Case row
Case 1
CurrentX = 1: CurrentY = 1
Print tens(1, 1); units(1, 1), tens(1, 2); units(1, 2), tens(1, 3); units(1, 3)
goal = 10 * (tens(1, 1) + tens(1, 2) + tens(1, 3))
goal = goal + units(1, 1) + units(1, 2) + units(1, 3)
Case 2
g = 10 * (tens(1, 1) + tens(1, 2) + tens(1, 3))
g = g + units(1, 1) + units(1, 2) + units(1, 3)
If g <> goal Then goodU = 0
Case 3
g = 10 * (tens(1, 1) + tens(1, 2) + tens(1, 3))
g = g + units(1, 1) + units(1, 2) + units(1, 3)
If g <> goal Then goodU = 0
If tens(3, 3) < tens(1, 1) Then goodU = 0
For c = 1 To 3
If goodU Then
g = 10 * (tens(1, c) + tens(2, c) + tens(3, c))
g = g + units(1, c) + units(2, c) + units(3, c)
If g <> goal Then goodU = 0: Exit For
End If
Next
End Select
End If
If goodU Then
If row = 3 And col = 3 Then
Open "double magic 2.txt" For Append As #2
For r = 1 To 3
For c = 1 To 3
Print tens(r, c); units(r, c),
Print #2, tens(r, c); units(r, c),
Next
Print: Print #2,
Next
ttl1 = 10 * (tens(1, 1) + tens(2, 2) + tens(3, 3))
ttl1 = ttl1 + units(1, 1) + units(2, 2) + units(3, 3)
ttl2 = 10 * (tens(3, 1) + tens(2, 2) + tens(1, 3))
ttl2 = ttl2 + units(3, 1) + units(2, 2) + units(1, 3)
Print goal; ttl1; ttl2: Print
Print #2, goal; ttl1; ttl2: Print #2,
Close 2
Else
r = row: c = col
c = c + 1: If c > 3 Then c = 1: r = r + 1
place r, c
End If
End If
usedU(u) = 0
End If
End If
Next u
usedT(t) = 0
End If
End If
Next t
End Sub
finds 168 sets where the sums of all the rows and columns are the same, and in fact 165. In no instance do both diagonals come out to 165. The number 168 does not include reflections and rotations, as the program requires that the top left number be the least (by checking its tens position), and that the top right number be lower than the bottom left.
A summary of the 168 such arrays shows 1) the common column or row total, 2) the major diagonal 3) the minor diagonal (i.e., top-right to bottom-left).
165 120 259
165 135 264
165 70 208
165 85 213
165 66 192
165 81 202
165 116 191
165 131 201
165 66 168
165 108 170
165 86 183
165 128 185
165 73 148
165 115 150
165 93 163
165 135 165
165 84 214
165 138 264
165 74 210
165 128 260
165 66 195
165 120 199
165 76 201
165 130 205
165 66 162
165 93 182
165 106 160
165 133 180
165 71 147
165 98 167
165 111 145
165 138 165
165 71 201
165 122 251
165 81 196
165 132 246
165 85 213
165 136 214
165 75 198
165 126 199
165 95 192
165 137 194
165 75 165
165 117 167
165 90 172
165 132 174
165 70 145
165 112 147
165 132 255
165 126 259
165 82 201
165 76 205
165 84 198
165 90 208
165 134 194
165 140 204
165 84 165
165 99 185
165 124 163
165 139 183
165 77 138
165 92 158
165 117 136
165 132 156
165 138 260
165 162 264
165 68 188
165 92 192
165 136 193
165 160 213
165 66 138
165 117 139
165 106 174
165 157 175
165 71 128
165 122 129
165 111 164
165 162 165
165 93 194
165 168 264
165 73 192
165 148 262
165 141 197
165 161 215
165 66 135
165 114 175
165 116 131
165 164 171
165 70 129
165 118 169
165 120 125
165 168 165
165 73 180
165 145 250
165 93 176
165 165 246
165 167 224
165 147 200
165 115 180
165 169 184
165 75 132
165 129 136
165 111 170
165 165 174
165 71 122
165 125 126
165 165 255
165 147 257
165 95 180
165 77 182
165 154 193
165 172 213
165 84 132
165 120 172
165 134 131
165 170 171
165 79 117
165 115 157
165 129 116
165 165 156
165 192 264
165 112 180
165 146 164
165 191 204
165 138 137
165 188 185
165 142 117
165 192 165
165 111 184
165 195 264
165 150 166
165 190 205
165 136 133
165 193 183
165 138 115
165 195 165
165 114 166
165 198 246
165 199 214
165 159 169
165 200 194
165 150 137
165 198 174
165 148 117
165 198 255
165 118 171
165 164 161
165 203 201
165 154 130
165 202 180
165 150 106
165 198 156
165 158 237
165 182 257
165 183 262
165 163 238
165 170 237
165 188 257
165 169 235
165 189 253
165 160 216
165 205 256
165 201 263
165 161 218
165 169 219
165 208 259
165 170 212
165 210 251
Note the following cases (among others) where there is a diagonal with the 165 total, and another that is close:
165 165 156
165 168 165
165 162 165
They are:
28 96 41
74 52 39
63 17 85
(165 165 156)
23 97 45
78 56 31
64 12 89
(165 168 165)
21 98 46
79 54 32
65 13 87
(165 162 165)
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Posted by Charlie
on 2009-10-16 18:59:07 |
computer solution
