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It's not a Magic Square (Posted on 2019-10-17) Difficulty: 4 of 5
In how many distinct ways can I fill a 3×3 grid with the integers 1 through 9 (each occurring exactly once) such that all cells in the grid are coprime with the neighbors? (Note that rotations and reflections count as distinct ways of filling the grid)

Note: A cell's neighbors are the cells that share a full side with it. If two cells share a corner but not a side, they are not neighbors.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution computer solution Comment 1 of 1
As indicated by the total at bottom of the output, there are 2016 total solutions. This includes rotations and reflections, so only 252 are shown below. The only rotation/reflection listed is the one in which the lowest corner is at the upper left and the top right corner is lower than the bottom left.

123
654
789

123
658
749

123
674
589

123
678
549

132
549
678

132
589
674

132
749
658

132
789
654

132
945
876

132
947
856

132
985
476

132
987
456

134
529
678

134
589
672

134
729
658

134
789
652

134
925
876

134
927
856

143
652
789

143
658
729

143
672
589

143
678
529

156
327
894

156
347
892

156
927
834

156
947
832

165
274
983

165
278
943

165
472
983

165
478
923

165
872
943

165
874
923

167
254
983

167
258
943

167
452
983

167
458
923

167
852
943

167
854
923

176
325
894

176
345
892

176
925
834

176
945
832

183
652
749

183
654
729

183
672
549

183
674
529

192
345
876

192
347
856

192
385
476

192
387
456

192
543
678

192
583
674

192
743
658

192
783
654

194
325
876

194
327
856

194
523
678

194
583
672

194
723
658

194
783
652

213
945
876

213
947
856

213
985
476

213
987
456

216
345
798

216
345
879

216
345
897

216
347
859

216
347
895

216
357
894

216
375
894

216
385
794

216
945
738

216
945
837

216
945
873

216
947
835

216
947
853

216
957
834

216
975
834

216
985
734

234
159
678

234
179
658

234
185
679

234
185
976

234
187
659

234
187
956

234
189
657

234
189
675

234
519
678

234
579
618

234
581
679

234
581
976

234
587
619

234
587
916

234
589
617

234
719
658

234
759
618

234
781
659

234
781
956

234
785
619

234
785
916

234
789
615

234
915
876

234
917
856

234
951
876

234
957
816

234
971
856

234
975
816

234
981
576

234
981
756

234
985
716

234
987
516

235
149
678

235
189
674

235
749
618

235
789
614

235
941
876

235
947
816

237
941
856

237
945
816

238
145
976

238
147
956

238
541
976

238
547
916

238
741
956

238
745
916

253
941
876

253
947
816

253
981
476

253
987
416

256
317
894

256
341
798

256
341
879

256
341
897

256
347
819

256
371
894

256
381
794

256
917
834

256
941
738

256
941
837

256
941
873

256
947
813

256
971
834

256
981
734

273
941
856

273
945
816

273
981
456

273
985
416

276
315
894

276
341
859

276
341
895

276
345
819

276
351
894

276
915
834

276
941
835

276
941
853

276
945
813

276
951
834

294
153
678

294
173
658

294
183
657

294
183
675

294
185
673

294
187
653

294
315
876

294
317
856

294
351
876

294
357
816

294
371
856

294
375
816

294
381
576

294
381
756

294
385
716

294
387
516

294
513
678

294
573
618

294
581
673

294
583
617

294
587
613

294
713
658

294
753
618

294
781
653

294
783
615

294
785
613

295
143
678

295
183
674

295
341
876

295
347
816

295
743
618

295
783
614

297
341
856

297
345
816

314
529
678

314
729
658

316
527
894

316
725
894

325
416
987

325
816
947

327
416
985

327
816
945

345
216
987

345
816
927

347
216
985

347
816
925

354
129
678

354
729
618

356
127
894

356
721
894

374
129
658

374
529
618

376
125
894

376
521
894

385
216
947

385
416
927

387
216
945

387
416
925

416
325
798

416
325
879

416
325
897

416
327
859

416
327
895

416
925
738

416
925
837

416
927
835

435
129
678

435
729
618

435
921
876

435
927
816

437
921
856

437
925
816

438
125
976

438
127
956

438
521
976

438
527
916

438
721
956

438
725
916

456
321
798

456
321
879

456
321
897

456
327
819

456
921
738

456
921
837

476
321
859

476
321
895

476
325
819

476
921
835

495
123
678

495
321
876

495
327
816

495
723
618

497
321
856

497
325
816

2016 done


DefDbl A-Z
Dim crlf$, sq(4, 4)

Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
  
 s$ = "123456789": h$ = s
 Do

   p = 0
   For r = 1 To 3
    For c = 1 To 3
      p = p + 1
      sq(r, c) = Val(Mid(s, p, 1))
    Next
   Next
   good = 1
   DoEvents
   For r = 1 To 3
    For c = 1 To 3
      For dr = 0 To 1
       For dc = 0 To 1
         If (dr <> 0 Or dc <> 0) And (dr = 0 Or dc = 0) Then
           r1 = r + dr: c1 = c + dc
           If sq(r1, c1) <> 0 Then
             If gcd(sq(r1, c1), sq(r, c)) > 1 Then good = 0: Exit For
           End If
         End If
       Next
       DoEvents
       If good = 0 Then Exit For
      Next
      If good = 0 Then Exit For
    Next
    If good = 0 Then Exit For
   Next
   If good Then
     If Mid(s, 1, 1) < Mid(s, 3, 1) And Mid(s, 1, 1) < Mid(s, 7, 1) And Mid(s, 1, 1) < Mid(s, 9, 1) Then
      If Mid(s, 3, 1) < Mid(s, 7, s) Then
       Text1.Text = Text1.Text & Left(s, 3) & crlf & Mid(s, 4, 3) & crlf & Right(s, 3) & crlf & crlf
      End If
     End If
     ct = ct + 1
   End If
   permute s
 Loop Until s = h
  
   
  Text1.Text = Text1.Text & ct & " done"
  DoEvents

End Sub

Function gcd(a, b)

  x = a: y = b
  Do
   q = Int(x / y)
   z = x - q * y
   x = y: y = z
  Loop Until z = 0
  gcd = x

End Function


  Posted by Charlie on 2019-10-17 13:53:36
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