+-----+-----+-----+-----+-----+-----+-----+
| EBF | EHG | EFG | EBE | AJD | EFJ | EIF |
+-----+-----+-----+-----+-----+-----+-----+
| ? | EFA | EIA | AJJ | EHC | EFI | EII |
+-----+-----+-----+-----+-----+-----+-----+
| EHD | EFD | ? | ACJ | EHB | EIH | EIH |
+-----+-----+-----+-----+-----+-----+-----+
| EFE | EIE | EBB | EHJ | ? | EBD | AJH |
+-----+-----+-----+-----+-----+-----+-----+
| EIJ | ? | AJB | EHI | EHG | EBG | EHA |
+-----+-----+-----+-----+-----+-----+-----+
| EIC | EBI | EHH | ? | EBA | AJG | EFC |
+-----+-----+-----+-----+-----+-----+-----+
| ? | ? | ? | ? | ? | ? | ? |
+-----+-----+-----+-----+-----+-----+-----+
- Each of the letters represents a different digit from 0 to 9. No number can contain any leading zero.
- J=0, G=E+A, B=C+I and, D=B-G
- Each of the columns add up to 1990. So does each of the rows and, each of the main diagonals.
Determine the seven 3-digit numbers missing from the bottom line.
Of the 88 sets of values for a - j that meet their given algebraic relationships, only one results in a top row total of 1990, marked with ** below:
a b c d e f g h i j row 1
1 8 2 3 4 7 5 9 6 0 2981
1 8 2 3 4 9 5 7 6 0 3005
1 8 6 3 4 7 5 9 2 0 2941
1 8 6 3 4 9 5 7 2 0 2965
1 9 2 3 5 4 6 8 7 0 3538
1 9 2 3 5 8 6 4 7 0 3586
1 9 2 5 3 6 4 8 7 0 2378
1 9 2 5 3 8 4 6 7 0 2402
1 9 4 2 6 3 7 8 5 0 4098
1 9 4 2 6 8 7 3 5 0 4158
1 9 4 6 2 7 3 8 5 0 1778
1 9 4 6 2 8 3 7 5 0 1790
1 9 5 2 6 3 7 8 4 0 4088
1 9 5 2 6 8 7 3 4 0 4148
1 9 5 6 2 7 3 8 4 0 1768
1 9 5 6 2 8 3 7 4 0 1780
1 9 7 3 5 4 6 8 2 0 3488
1 9 7 3 5 8 6 4 2 0 3536
1 9 7 5 3 6 4 8 2 0 2328
1 9 7 5 3 8 4 6 2 0 2352
2 9 1 3 4 5 6 7 8 0 3059
2 9 1 3 4 7 6 5 8 0 3083
2 9 1 4 3 6 5 7 8 0 2479
2 9 1 4 3 7 5 6 8 0 2491
2 9 4 1 6 3 8 7 5 0 4189
2 9 4 1 6 7 8 3 5 0 4237
2 9 4 6 1 7 3 8 5 0 1277
2 9 4 6 1 8 3 7 5 0 1289
2 9 5 1 6 3 8 7 4 0 4179
2 9 5 1 6 7 8 3 4 0 4227
2 9 5 6 1 7 3 8 4 0 1267
2 9 5 6 1 8 3 7 4 0 1279
2 9 8 3 4 5 6 7 1 0 2989
2 9 8 3 4 7 6 5 1 0 3013
2 9 8 4 3 6 5 7 1 0 2409
2 9 8 4 3 7 5 6 1 0 2421
3 8 2 1 4 5 7 9 6 0 3139
3 8 2 1 4 9 7 5 6 0 3187
3 8 6 1 4 5 7 9 2 0 3099
3 8 6 1 4 9 7 5 2 0 3147
3 9 1 2 4 5 7 6 8 0 3150
3 9 1 2 4 6 7 5 8 0 3162
3 9 1 4 2 6 5 7 8 0 1978
3 9 1 4 2 7 5 6 8 0 1990 **
3 9 2 1 5 4 8 6 7 0 3720
3 9 2 1 5 6 8 4 7 0 3744
3 9 2 5 1 6 4 8 7 0 1376
3 9 2 5 1 8 4 6 7 0 1400
3 9 7 1 5 4 8 6 2 0 3670
3 9 7 1 5 6 8 4 2 0 3694
3 9 7 5 1 6 4 8 2 0 1326
3 9 7 5 1 8 4 6 2 0 1350
3 9 8 2 4 5 7 6 1 0 3080
3 9 8 2 4 6 7 5 1 0 3092
3 9 8 4 2 6 5 7 1 0 1908
3 9 8 4 2 7 5 6 1 0 1920
4 8 2 1 3 5 7 9 6 0 2638
4 8 2 1 3 9 7 5 6 0 2686
4 8 2 3 1 7 5 9 6 0 1478
4 8 2 3 1 9 5 7 6 0 1502
4 8 6 1 3 5 7 9 2 0 2598
4 8 6 1 3 9 7 5 2 0 2646
4 8 6 3 1 7 5 9 2 0 1438
4 8 6 3 1 9 5 7 2 0 1462
4 9 1 2 3 5 7 6 8 0 2649
4 9 1 2 3 6 7 5 8 0 2661
4 9 1 3 2 5 6 7 8 0 2057
4 9 1 3 2 7 6 5 8 0 2081
4 9 8 2 3 5 7 6 1 0 2579
4 9 8 2 3 6 7 5 1 0 2591
4 9 8 3 2 5 6 7 1 0 1987
4 9 8 3 2 7 6 5 1 0 2011
5 9 2 1 3 4 8 6 7 0 2718
5 9 2 1 3 6 8 4 7 0 2742
5 9 2 3 1 4 6 8 7 0 1534
5 9 2 3 1 8 6 4 7 0 1582
5 9 7 1 3 4 8 6 2 0 2668
5 9 7 1 3 6 8 4 2 0 2692
5 9 7 3 1 4 6 8 2 0 1484
5 9 7 3 1 8 6 4 2 0 1532
6 9 4 1 2 3 8 7 5 0 2185
6 9 4 1 2 7 8 3 5 0 2233
6 9 4 2 1 3 7 8 5 0 1593
6 9 4 2 1 8 7 3 5 0 1653
6 9 5 1 2 3 8 7 4 0 2175
6 9 5 1 2 7 8 3 4 0 2223
6 9 5 2 1 3 7 8 4 0 1583
6 9 5 2 1 8 7 3 4 0 1643
The table then looks like:
+-----+-----+-----+-----+-----+-----+-----+
| 297 | 265 | 275 | 292 | 304 | 270 | 287 | 1990
+-----+-----+-----+-----+-----+-----+-----+
| ? | 273 | 283 | 300 | 261 | 278 | 288 | 1683
+-----+-----+-----+-----+-----+-----+-----+
| 264 | 274 | ? | 310 | 269 | 286 | 286 | 1689
+-----+-----+-----+-----+-----+-----+-----+
| 272 | 282 | 299 | 260 | ? | 294 | 306 | 1713
+-----+-----+-----+-----+-----+-----+-----+
| 280 | ? | 309 | 268 | 265 | 295 | 263 | 1680
+-----+-----+-----+-----+-----+-----+-----+
| 281 | 298 | 266 | ? | 293 | 305 | 271 | 1714
+-----+-----+-----+-----+-----+-----+-----+
| ? | ? | ? | ? | ? | ? | ? |
+-----+-----+-----+-----+-----+-----+-----+
1394 1392 1432 1430 1392 1728 1701
The ?'s in the first six rows can be filled in by completing 1990 from the totals of the other entries, given at the right. Then the 1990's for the rows can be completed using the ?'s at the bottom, which are therefore forced.
+-----+-----+-----+-----+-----+-----+-----+
| 297 | 265 | 275 | 292 | 304 | 270 | 287 |
+-----+-----+-----+-----+-----+-----+-----+
| 307+| 273 | 283 | 300 | 261 | 278 | 288 |
+-----+-----+-----+-----+-----+-----+-----+
| 264 | 274 | 301+| 310 | 269 | 286 | 286 |
+-----+-----+-----+-----+-----+-----+-----+
| 272 | 282 | 299 | 260 | 277+| 294 | 306 |
+-----+-----+-----+-----+-----+-----+-----+
| 280 | 310+| 309 | 268 | 265 | 295 | 263 |
+-----+-----+-----+-----+-----+-----+-----+
| 281 | 298 | 266 | 276+| 293 | 305 | 271 |
+-----+-----+-----+-----+-----+-----+-----+
| ? | ? | ? | ? | ? | ? | ? |
+-----+-----+-----+-----+-----+-----+-----+
1394 1392 1432 1430 1392 1728 1701
307 310 301 276 277
---- ---- ---- ---- ----
1701 1702 1733 1706 1669
+-----+-----+-----+-----+-----+-----+-----+
| 297 | 265 | 275 | 292 | 304 | 270 | 287 |
+-----+-----+-----+-----+-----+-----+-----+
| 307+| 273 | 283 | 300 | 261 | 278 | 288 |
+-----+-----+-----+-----+-----+-----+-----+
| 264 | 274 | 301+| 310 | 269 | 286 | 286 |
+-----+-----+-----+-----+-----+-----+-----+
| 272 | 282 | 299 | 260 | 277+| 294 | 306 |
+-----+-----+-----+-----+-----+-----+-----+
| 280 | 310+| 309 | 268 | 265 | 295 | 263 |
+-----+-----+-----+-----+-----+-----+-----+
| 281 | 298 | 266 | 276+| 293 | 305 | 271 |
+-----+-----+-----+-----+-----+-----+-----+
| 289 | 288 | 257 | 284 | 321 | 262 | 289 |
+-----+-----+-----+-----+-----+-----+-----+
The ?'d entries marked now with + to indicate their values had to be added to the previous column values before complementing re 1990 to get the bottom line.
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Posted by Charlie
on 2013-08-05 21:23:51 |
computer-assisted solution
