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Try to solve kakuro with the principle of exclusion. By examining the block length and the sum of the bordering black field you can see that there can be only a fixed set of solutions. We prepared the tables with possible combinations of sums by the block length.

Tables with possible sums:

with field number: 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9

with 2 fields:
sum ( reference number ) : possible solution
03: 12
04: 13
05: 14 - 23
06: 15 - 24
07: 16 - 25 - 34
08: 17 - 26 - 35
09: 18 - 27 - 36 - 45
10: 19 - 28 - 37 - 46
11: 29 - 38 - 47 - 56
12: 39 - 48 - 57
13: 49 - 58 - 67
14: 59 - 68
15: 69 - 78
16: 79
17: 89



with 3 fields: TOP
sum ( reference number ) : possible solution
06: 123
07: 124
08: 125 - 134
09: 126 - 135 - 234
10: 127 - 136 - 145 - 235
11: 128 - 137 - 146 - 236 - 245
12: 129 - 138 - 147 - 156 - 237 - 246 - 345
13: 139 - 148 - 157 - 238 - 247 - 256 - 346
14: 149 - 158 - 167 - 239 - 248 - 257 - 347 - 356
15: 159 - 168 - 249 - 258 - 267 - 348 - 357 - 456
16: 169 - 178 - 259 - 268 - 349 - 358 - 367 - 457
17: 179 - 269 - 278 - 359 - 368 - 458 - 467
18: 189 - 279 - 369 - 378 - 459 - 468 - 567
19: 289 - 379 - 469 - 478 - 568
20: 389 - 479 - 569 - 578
21: 489 - 579 - 678
22: 589 - 679
23: 689
24: 789



with 4 fields: TOP
sum ( reference number ) : possible solution
10: 1234
11: 1235
12: 1236 - 1245
13: 1237 - 1246 - 1345
14: 1238 - 1247 - 1256 - 1346 - 2345
15: 1239 - 1248 - 1257 - 1347 - 1356 - 2346
16: 1249 - 1258 - 1267 - 1348 - 1357 - 1456 - 2347 - 2356
17: 1259 - 1268 - 1349 - 1358 - 1367 - 1457 - 2348 - 2357 - 2456
18: 1269 - 1278 - 1359 - 1368 - 1458 - 1467 - 2349 - 2358 - 2367 - 2457 - 3456
19: 1279 - 1369 - 1378 - 1459 - 1468 - 1567 - 2359 - 2368 - 2458 - 2467 - 3457
20: 1289 - 1379 - 1469 - 1478 - 1568 - 2369 - 2378 - 2459 - 2468 - 2567 - 3458 - 3467
21: 1389 - 1479 - 1569 - 1578 - 2379 - 2469 - 2478 - 2568 - 3459 - 3468 - 3567
22: 1489 - 1579 - 1678 - 2389 - 2479 - 2569 - 2578 - 3469 - 3478 - 3568 - 4567
23: 1589 - 1679 - 2489 - 2579 - 2678 - 3479 - 3569 - 3578 - 4568
24: 1689 - 2589 - 2679 - 3489 - 3579 - 3678 - 4569 - 4578
25: 1789 - 2689 - 3589 - 3679 - 4579 - 4678
26: 2789 - 3689 - 4589 - 4679 - 5678
27: 3789 - 4689 - 5679
28: 4789 - 5689
29: 5789
30: 6789



with 5 fields: TOP
sum ( reference number ) : possible solution
15: 12345
16: 12346
17: 12347 - 12356
18: 12348 - 12357 - 12456
19: 12349 - 12358 - 12367 - 12457 - 13456
20: 12359 - 12368 - 12458 - 12467 - 13457 - 23456
21: 12369 - 12378 - 12459 - 12468 - 12567 - 13458 - 13467 - 23457
22: 12379 - 12469 - 12478 - 12568 - 13459 - 13468 - 13567 - 23458 - 23467
23: 12389 - 12479 - 12569 - 12578 - 13469 - 13478 - 13568 - 14567 - 23459 - 23468 - 23567
24: 12489 - 12579 - 12678 - 13479 - 13569 - 13578 - 14568 - 23469 - 23478 - 23568 - 24567
25: 12589 - 12679 - 13489 - 13579 - 13678 - 14569 - 14578 - 23479 - 23569 - 23578 - 24568 - 34567
26: 12689 - 13589 - 13679 - 14579 - 14678 - 23489 - 23579 - 23678 - 24569 - 24578 - 34568
27: 12789 - 13689 - 14589 - 14679 - 15678 - 23589 - 23679 - 24579 - 24678 - 34569 - 34578
28: 13789 - 14689 - 15679 - 23689 - 24589 - 24679 - 25678 - 34579 - 34678
29: 14789 - 15689 - 23789 - 24689 - 25679 - 34589 - 34679 - 35678
30: 15789 - 24789 - 25689 - 34689 - 35679 - 45678
31: 16789 - 25789 - 34789 - 35689 - 45679
32: 26789 - 35789 - 45689
33: 36789 - 45789
34: 46789
35: 56789



with 6 fields: TOP
sum ( reference number ) : possible solution
21: 123456
22: 123457
23: 123458 - 123467
24: 123459 - 123468 - 123567
25: 123469 - 123478 - 123568 - 124567
26: 123479 - 123569 - 123578 - 124568 - 134567
27: 123489 - 123579 - 123678 - 124569 - 124578 - 134568 - 234567
28: 123589 - 123679 - 124579 - 124678 - 134569 - 134578 - 234568
29: 123689 - 124589 - 124679 - 125678 - 134579 - 134678 - 234569 - 234578
30: 123789 - 124689 - 125679 - 134589 - 134679 - 135678 - 234579 - 234678
31: 124789 - 125689 - 134689 - 135679 - 145678 - 234589 - 234679 - 235678
32: 125789 - 134789 - 135689 - 145679 - 234689 - 235679 - 245678
33: 126789 - 135789 - 145689 - 234789 - 235689 - 245679 - 345678
34: 136789 - 145789 - 235789 - 245689 - 345679
35: 146789 - 236789 - 245789 - 345689
36: 156789 - 246789 - 345789
37: 256789 - 346789
38: 356789
39: 456789



with 7 fields: TOP
sum ( reference number ) : possible solution
28: 1234567
29: 1234568
30: 1234569 - 1234578
31: 1234579 - 1234678
32: 1234589 - 1234679 - 1235678
33: 1234689 - 1235679 - 1245678
34: 1234789 - 1235689 - 1245679 - 1345678
35: 1235789 - 1245689 - 1345679 - 2345678
36: 1236789 - 1245789 - 1345689 - 2345679
37: 1246789 - 1345789 - 2345689
38: 1256789 - 1346789 - 2345789
39: 1356789 - 2346789
40: 1456789 - 2356789
41: 2456789
42: 3456789



with 8 fields: TOP
sum ( reference number ) : possible solution
36: 12345678
37: 12345679
38: 12345689
39: 12345789
40: 12346789
41: 12356789
42: 12456789
43: 13456789
44: 23456789



with 9 fields: TOP
sum ( reference number ) : possible solution
45: 123456789