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 Double the Magic (Posted on 2009-10-16)
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?

 See The Solution Submitted by brianjn No Rating

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 computer solution | Comment 1 of 2

Dim tens(3, 3), units(3, 3), goal
Dim usedT(9), usedU(9)

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)

 Posted by Charlie on 2009-10-16 18:59:07

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