Killer Sudoku Combinations Chart

Sudoku69.com

This Killer Sudoku combinations chart lists every set of digits that can fill a cage, for every cage size from 2 to 9 cells and every possible sum. Find the cage size, find its sum, and read off which digits it can hold. It works as a Killer Sudoku cheat sheet for beginners and intermediate solvers alike: keep it open next to the grid.

Digits 1 to 9 never repeat inside a cage, so a cage of n cells is simply n different digits that add up to the sum. There are 502 such digit sets across cage sizes 2 to 9. Rows with a single combination are highlighted, because those cages tell you their digits straight away.

How do you use the chart mid-solve?

Rule out digits. Look up the cage and cross off every digit that appears in no listed combination. A 3-cell cage of 8 is 1,2,5 or 1,3,4, so 6, 7, 8 and 9 cannot go in it.
Spot single-combination cages. A highlighted row means the digit set is fixed. Write those digits as the only candidates and erase the rest. The cage combinations technique explains what to do with them next.
Narrow as you go. Once a digit is placed elsewhere in the cage's row, column or box, cross out every combination containing it. Many cages drop to one combination this way.
Use complements. An n-cell cage summing to S uses exactly the digits missing from the (9-n)-cell set summing to 45-S. A 7-cell cage of 28 is the reverse of a 2-cell sum of 17, which is 8 and 9, so the cage holds 1 to 7. This is the same idea behind innies and outies, where a whole row, column or box adds up to 45.

The chart

2-cell cages

Sum	Combinations	Count
3	12	1
4	13	1
5	2314	2
6	2415	2
7	342516	3
8	352617	3
9	45362718	4
10	46372819	4
11	56473829	4
12	574839	3
13	675849	3
14	6859	2
15	7869	2
16	79	1
17	89	1

3-cell cages

Sum	Combinations	Count
6	123	1
7	124	1
8	134125	2
9	234135126	3
10	235145136127	4
11	245236146137128	5
12	345246156237147138129	7
13	346256247157238148139	7
14	356347257167248158239149	8
15	456357267348258168249159	8
16	457367358268178349259169	8
17	467458368278359269179	7
18	567468378459369279189	7
19	568478469379289	5
20	578569479389	4
21	678579489	3
22	679589	2
23	689	1
24	789	1

4-cell cages

Sum	Combinations	Count
10	1234	1
11	1235	1
12	12451236	2
13	134512461237	3
14	23451346125612471238	5
15	234613561347125712481239	6
16	23561456234713571267134812581249	8
17	245623571457136723481358126813491259	9
18	34562457236714672358145813681278234913591269	11
19	34572467156724582368146813782359145913691279	11
20	346725673458246815682378147824592369146913791289	12
21	35673468256824781578345924691569237914791389	11
22	45673568347825781678346925692479157923891489	11
23	456835782678356934792579167924891589	9
24	45783678456935792679348925891689	8
25	467845793679358926891789	6
26	56784679458936892789	5
27	567946893789	3
28	56894789	2
29	5789	1
30	6789	1

5-cell cages

Sum	Combinations	Count
15	12345	1
16	12346	1
17	1235612347	2
18	124561235712348	3
19	1345612457123671235812349	5
20	234561345712467124581236812359	6
21	2345713467125671345812468123781245912369	8
22	234671356723458134681256812478134591246912379	9
23	2356714567234681356813478125782345913469125691247912389	11
24	2456723568145682347813578126782346913569134791257912489	11
25	345672456823578145781367823569145692347913579126791348912589	12
26	3456824578236781467824569235791457913679234891358912689	11
27	3457824678156783456924579236791467923589145891368912789	11
28	346782567834579246791567924589236891468913789	9
29	3567834679256793458924689156892378914789	8
30	456783567934689256892478915789	6
31	4567935689347892578916789	5
32	456893578926789	3
33	4578936789	2
34	46789	1
35	56789	1

6-cell cages

Sum	Combinations	Count
21	123456	1
22	123457	1
23	123467123458	2
24	123567123468123459	3
25	124567123568123478123469	4
26	134567124568123578123569123479	5
27	234567134568124578123678124569123579123489	7
28	234568134578124678134569124579123679123589	7
29	234578134678125678234569134579124679124589123689	8
30	234678135678234579134679125679134589124689123789	8
31	235678145678234679135679234589134689125689124789	8
32	245678235679145679234689135689134789125789	7
33	345678245679235689145689234789135789126789	7
34	345679245689235789145789136789	5
35	345689245789236789146789	4
36	345789246789156789	3
37	346789256789	2
38	356789	1
39	456789	1

7-cell cages

Sum	Combinations	Count
28	1234567	1
29	1234568	1
30	12345781234569	2
31	12346781234579	2
32	123567812346791234589	3
33	124567812356791234689	3
34	1345678124567912356891234789	4
35	2345678134567912456891235789	4
36	2345679134568912457891236789	4
37	234568913457891246789	3
38	234578913467891256789	3
39	23467891356789	2
40	23567891456789	2
41	2456789	1
42	3456789	1

8-cell cages

Sum	Combinations	Count
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

9-cell cages

Sum	Combinations	Count
45	123456789	1

Which sums are worth memorising?

Only the extreme sums have one combination. For 2-cell cages these are 3, 4, 16 and 17. The same four positions, the two lowest and the two highest possible sums, are unique for cages of 3 to 7 cells too. An 8-cell cage always has one combination (any sum from 36 to 44), and a 9-cell cage is always 1 to 9, summing to 45.

Learn the 2, 3 and 4-cell rows first. They turn up on every grid, and the forced sums article shows them in play. If you are new to cages and sums, start with how to play Killer Sudoku, then practise on easy Killer Sudoku with the chart beside you.

Killer Sudoku Cage Combinations How to Play Killer Sudoku Innies and Outies in Killer Sudoku Killer Sudoku and Math Variants