Kakuro

Kakuro or Kakkuro is a mathematical crossword puzzle. The puzzle is played on a grid of black and white cells with clues given in triangle shaped cells scattered throughout the grid. The objective of the puzzle is to insert a digit from 1 to 9 into each white cell such that the sum of the numbers in each entry matches the clue associated with it. The clue for each vertical entry is the triangle cell above that entry. The clue for each horizontal entry is the triangle cell to the left of that entry. One additional rule is that no digit is duplicated in any entry.

The top left clue in the above example says "23" which means that the three empty cells in this column must add up to 23. In this case, the correct answer is 9+8+6=23. In the bottom left section of the example, you will see a triangle clue that says "6" which points to 3 empty cells in a row. The only way to fill this without repeating numbers is 1+2+3=6.

Kakuro Solving Techniques

There are certain sums that can only be attained with small sets of numbers. For example, a sum of 4 with two cells can only be filled with 1+3. A sum of 34 with five cells must be 4+6+7+8+9. Looking for these spots in the puzzle is a good starting point. Use light pencil marks to note all the possible numbers that could fit into each cell. By comparing the horizontal and vertical clues, you can often eliminate the possibilities down to one number.

When you get stuck, try finding a square where there are only a few possibilities and make a guess. Follow through this guess until you reach an impossible condition. Now you can eliminate that guess. If you don't reach an impossible condition, that might be the right answer, but you should try the other guesses to make sure.

To help you get started, here is a cheat sheet of all the possible sums:

2 Spaces
3: 12
4: 13
5: 14, 23
6: 15, 24
7: 16, 25, 34
8: 17, 26, 35
9: 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

3 Spaces
6: 123
7: 124
8: 125, 134
9: 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

4 Spaces
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

5 Spaces
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

6 Spaces
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

7 Spaces
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

8 Spaces
36: 12345678
37: 12345679
38: 12345689
39: 12345789
40: 12346789
41: 12356789
42: 12456789
43: 13456789
44: 23456789

9 Spaces
45: 123456789