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Killer Sudoku Cage Combinations

This is the whole table, computed by the same code that builds the puzzles rather than copied from anywhere. For each cage size, every total it can carry and every set of digits that reaches it. Thirty-four of those totals have only one answer, and those are the ones worth memorising.

How to read it

What people call a killer sudoku combinations chart is the answer to one question asked 120 times: which sets of different digits from 1 to 9 add up to this total? A cage of n cells holds n different digits, so its total is pinned between two extremes. The smallest is the first n digits added together and the largest is the last n, which puts a four-cell cage between 10, being 1+2+3+4, and 30, being 6+7+8+9. A total outside that range cannot belong to a cage of that size, and that is the cheapest error check there is when you are copying a board off paper.

Of those 120 totals, 34 admit exactly one set of digits. They are marked in the tables below, and they are the cages that place their contents the moment you read them.

The table is a mirror

Look down the two-cell list and it reads the same forwards and backwards: one combination at 3 and one at 17, two at 5 and two at 15, four at 9 and four at 11. That is not a coincidence, and understanding why halves what there is to remember.

Swap every digit d for 10 minus d, and a set of different digits from 1 to 9 turns into another set of different digits from 1 to 9. A cage of n cells totalling T becomes a cage of n cells totalling 10n minus T. So the fillings of a five-cell cage totalling 20 correspond one for one with those of a five-cell cage totalling 30. Learn the bottom half of any column and you have been handed the top half.

Cages of eight and nine need no table at all

Eight different digits out of nine means exactly one is missing, and since the whole set adds to 45, the missing digit is 45 minus the total. A cage of eight cells totalling 40 contains everything except the 5. Every eight-cell total therefore has precisely one answer, which is why that stretch of the table is a straight run of nine single entries. A nine-cell cage totals 45 and holds all nine digits, and there is nothing at all to work out.

Every cage total, and what can fill it

Cages of 2 cells

Totals run from 3 to 17, and four of them – 3, 4, 16 and 17 – can be filled only one way. Those four are marked below. A forced total names the pair of digits and never which of the two cells takes which; the row, column and box rules settle that afterwards.

Total Possible digits Ways
3 12 1
4 13 1
5 23 14 2
6 24 15 2
7 34 25 16 3
8 35 26 17 3
9 45 36 27 18 4
10 46 37 28 19 4
11 56 47 38 29 4
12 57 48 39 3
13 67 58 49 3
14 68 59 2
15 78 69 2
16 79 1
17 89 1

Cages of 3 cells

Totals run from 6 to 24, and four of them – 6, 7, 23 and 24 – can be filled only one way. Those four are marked below.

Total Possible digits Ways
6 123 1
7 124 1
8 134 125 2
9 234 135 126 3
10 235 145 136 127 4
11 245 236 146 137 128 5
12 345 246 156 237 147 138 129 7
13 346 256 247 157 238 148 139 7
14 356 347 257 167 248 158 239 149 8
15 456 357 267 348 258 168 249 159 8
16 457 367 358 268 178 349 259 169 8
17 467 458 368 278 359 269 179 7
18 567 468 378 459 369 279 189 7
19 568 478 469 379 289 5
20 578 569 479 389 4
21 678 579 489 3
22 679 589 2
23 689 1
24 789 1

Cages of 4 cells

Totals run from 10 to 30, and four of them – 10, 11, 29 and 30 – can be filled only one way. Those four are marked below.

Total Possible digits Ways
10 1234 1
11 1235 1
12 1245 1236 2
13 1345 1246 1237 3
14 2345 1346 1256 1247 1238 5
15 2346 1356 1347 1257 1248 1239 6
16 2356 1456 2347 1357 1267 1348 1258 1249 8
17 2456 2357 1457 1367 2348 1358 1268 1349 1259 9
18 3456 2457 2367 1467 2358 1458 1368 1278 2349 1359 1269 11
19 3457 2467 1567 2458 2368 1468 1378 2359 1459 1369 1279 11
20 3467 2567 3458 2468 1568 2378 1478 2459 2369 1469 1379 1289 12
21 3567 3468 2568 2478 1578 3459 2469 1569 2379 1479 1389 11
22 4567 3568 3478 2578 1678 3469 2569 2479 1579 2389 1489 11
23 4568 3578 2678 3569 3479 2579 1679 2489 1589 9
24 4578 3678 4569 3579 2679 3489 2589 1689 8
25 4678 4579 3679 3589 2689 1789 6
26 5678 4679 4589 3689 2789 5
27 5679 4689 3789 3
28 5689 4789 2
29 5789 1
30 6789 1

Cages of 5 cells

Totals run from 15 to 35, and four of them – 15, 16, 34 and 35 – can be filled only one way. Those four are marked below.

Total Possible digits Ways
15 12345 1
16 12346 1
17 12356 12347 2
18 12456 12357 12348 3
19 13456 12457 12367 12358 12349 5
20 23456 13457 12467 12458 12368 12359 6
21 23457 13467 12567 13458 12468 12378 12459 12369 8
22 23467 13567 23458 13468 12568 12478 13459 12469 12379 9
23 23567 14567 23468 13568 13478 12578 23459 13469 12569 12479 12389 11
24 24567 23568 14568 23478 13578 12678 23469 13569 13479 12579 12489 11
25 34567 24568 23578 14578 13678 23569 14569 23479 13579 12679 13489 12589 12
26 34568 24578 23678 14678 24569 23579 14579 13679 23489 13589 12689 11
27 34578 24678 15678 34569 24579 23679 14679 23589 14589 13689 12789 11
28 34678 25678 34579 24679 15679 24589 23689 14689 13789 9
29 35678 34679 25679 34589 24689 15689 23789 14789 8
30 45678 35679 34689 25689 24789 15789 6
31 45679 35689 34789 25789 16789 5
32 45689 35789 26789 3
33 45789 36789 2
34 46789 1
35 56789 1

Cages of 6 cells

Totals run from 21 to 39, and four of them – 21, 22, 38 and 39 – can be filled only one way. Those four are marked below.

Total Possible digits Ways
21 123456 1
22 123457 1
23 123467 123458 2
24 123567 123468 123459 3
25 124567 123568 123478 123469 4
26 134567 124568 123578 123569 123479 5
27 234567 134568 124578 123678 124569 123579 123489 7
28 234568 134578 124678 134569 124579 123679 123589 7
29 234578 134678 125678 234569 134579 124679 124589 123689 8
30 234678 135678 234579 134679 125679 134589 124689 123789 8
31 235678 145678 234679 135679 234589 134689 125689 124789 8
32 245678 235679 145679 234689 135689 134789 125789 7
33 345678 245679 235689 145689 234789 135789 126789 7
34 345679 245689 235789 145789 136789 5
35 345689 245789 236789 146789 4
36 345789 246789 156789 3
37 346789 256789 2
38 356789 1
39 456789 1

Cages of 7 cells

Totals run from 28 to 42, and four of them – 28, 29, 41 and 42 – can be filled only one way. Those four are marked below.

Total Possible digits Ways
28 1234567 1
29 1234568 1
30 1234578 1234569 2
31 1234678 1234579 2
32 1235678 1234679 1234589 3
33 1245678 1235679 1234689 3
34 1345678 1245679 1235689 1234789 4
35 2345678 1345679 1245689 1235789 4
36 2345679 1345689 1245789 1236789 4
37 2345689 1345789 1246789 3
38 2345789 1346789 1256789 3
39 2346789 1356789 2
40 2356789 1456789 2
41 2456789 1
42 3456789 1

Cages of 8 cells

Totals run from 36 to 44, and every one of the nine can be filled only one way, which is why the whole of this table is marked. Eight different digits leave exactly one of the nine out, and 45 minus the total names it.

Total Possible digits Ways
36 12345678 1
37 12345679 1
38 12345689 1
39 12345789 1
40 12346789 1
41 12356789 1
42 12456789 1
43 13456789 1
44 23456789 1

Cages of 9 cells

45 is the only total a nine-cell cage can carry, and all nine digits are the only filling. It is marked below like the other forced totals, and it is the one of the 34 that narrows nothing at all.

Total Possible digits Ways
45 123456789 1

Using it without memorising it

Nobody learns 120 rows. What experienced solvers actually carry is the four forced two-cell totals, at 3, 4, 16 and 17; the four forced three-cell ones, at 6, 7, 23 and 24; and the habit of working from the extremes. The lowest few totals for any size can only use the smallest digits and the highest few only the largest, which constrains cells without needing the exact combination. A five-cell cage totalling 17 must contain both a 1 and a 2 whatever else it holds, because without them five different digits cannot reach down that far.

The middle of the table is where to stop trying. A four-cell cage totalling 20 has twelve possible fillings, and a five-cell cage totalling 25 has twelve as well; those are the worst cases anywhere in the table. Cages like that are not where a puzzle opens up, and time spent writing out their options is usually time lost.

Which is the honest limit of a killer sudoku cage combinations table. It answers a question about arithmetic in isolation, and a cage is never in isolation: it sits across rows, columns and boxes that have already ruled digits out. The table narrows the cage, the grid narrows it further, and the second step is usually the bigger one.