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Strategies for Popular Number Puzzles

KenKen Combinations

KenKen cage combinations are a whole different beast to those you may be used to with Killer Sudoku. That is because the Killer Convention does not apply. Cages can have duplicate numbers provided they do not duplicate in the rows and columns. If a cage is all in a line then the duplication doesn't arise. But if there is any kind of kink - what I call a "dog leg", then we have to include duplicates. But how many dog-legs can there be in a KenKen cage. Normal 6x6 KenKen has a cap of 6 cells in a cage although these are rare. Two cell cages must be adjacent and therefore can never have a dog-leg. So from three to six cells we need a combinations table for each of the number of dog-legs.

See the article on KenDoku Dog Legs for a distinction between the two types of puzzles.

The tables below are separated into three. These are the combinations for no dog-legs, one dog-leg and two. Beyond that the solver will compute the combinations on the fly.

I have created this diagram which shows all the possible cage shapes with the maximum number of duplicate positions. These are labelled A and B - since some cages we can have more than one duplicate solution number.
KenKen Cage dog legs

Straight Line 6x6 Cage Combos

Cage SizeSpread+×-÷
21-23 + 2 ×1 -2 ÷Unique
21-34 + 3 ×2 -3 ÷Unique
21-45 + 4 ×3 -4 ÷
22-35 + 6 ×1 -
21-56 + 5 ×4 -5 ÷
22-46 + 8 ×2 -2 ÷
21-67 + 6 ×5 -6 ÷
22-57 + 10 ×3 -
23-47 + 12 ×1 -
22-68 + 12 ×4 -3 ÷
23-58 + 15 ×2 -
23-69 + 18 ×3 -2 ÷
24-59 + 20 ×1 -
24-610 + 24 ×2 -Unique
25-611 + 30 ×1 -Unique
31/2/36 + 6 ×Unique
31/2/47 + 8 ×Unique
31/2/58 + 10 ×
31/3/48 + 12 ×
31/2/69 + 12 ×
31/3/59 + 15 ×
32/3/49 + 24 ×
31/3/610 + 18 ×
31/4/510 + 20 ×
32/3/510 + 30 ×
31/4/611 + 24 ×
32/3/611 + 36 ×
32/4/511 + 40 ×
31/5/612 + 30 ×
32/4/612 + 48 ×
33/4/512 + 60 ×
32/5/613 + 60 ×
33/4/613 + 72 ×
33/5/614 + 90 ×Unique
34/5/615 + 120 ×Unique
41/2/3/410 + 24 ×Unique
41/2/3/511 + 30 ×Unique
41/2/3/612 + 36 ×
41/2/4/512 + 40 ×
41/2/4/613 + 48 ×
41/3/4/513 + 60 ×
41/2/5/614 + 60 ×
41/3/4/614 + 72 ×
42/3/4/514 + 120 ×
41/3/5/615 + 90 ×
42/3/4/615 + 144 ×
41/4/5/616 + 120 ×
42/3/5/616 + 180 ×
42/4/5/617 + 240 ×Unique
43/4/5/618 + 360 ×Unique
51/2/3/4/515 + 120 ×Unique
51/2/3/4/616 + 144 ×Unique
51/2/3/5/617 + 180 ×Unique
51/2/4/5/618 + 240 ×Unique
51/3/4/5/619 + 360 ×Unique
52/3/4/5/620 + 720 ×Unique

One dog-leg 6x6 Cage Combos

Cage SizeSpread+×-÷
31/1/24 + 2 ×Unique
31/1/35 + 3 ×
31/2/25 + 4 ×
31/1/46 + 4 ×
31/2/36 + 6 ×
31/1/57 + 5 ×
31/2/47 + 8 ×
31/3/37 + 9 ×
32/2/37 + 12 ×
31/1/68 + 6 ×
31/2/58 + 10 ×
31/3/48 + 12 ×
32/2/48 + 16 ×
32/3/38 + 18 ×
31/2/69 + 12 ×
31/3/59 + 15 ×
31/4/49 + 16 ×
32/2/59 + 20 ×
32/3/49 + 24 ×
31/3/610 + 18 ×
31/4/510 + 20 ×
32/2/610 + 24 ×
32/3/510 + 30 ×
32/4/410 + 32 ×
33/3/410 + 36 ×
31/4/611 + 24 ×
31/5/511 + 25 ×
32/3/611 + 36 ×
32/4/511 + 40 ×
33/3/511 + 45 ×
33/4/411 + 48 ×
31/5/612 + 30 ×
32/4/612 + 48 ×
32/5/512 + 50 ×
33/3/612 + 54 ×
33/4/512 + 60 ×
31/6/613 + 36 ×
32/5/613 + 60 ×
33/4/613 + 72 ×
33/5/513 + 75 ×
34/4/513 + 80 ×
32/6/614 + 72 ×
33/5/614 + 90 ×
34/4/614 + 96 ×
34/5/514 + 100 ×
33/6/615 + 108 ×
34/5/615 + 120 ×
34/6/616 + 144 ×
35/5/616 + 150 ×
35/6/617 + 180 ×Unique
41/1/2/37 + 6 ×Unique
41/1/2/48 + 8 ×
41/2/2/38 + 12 ×
41/1/2/59 + 10 ×
41/1/3/49 + 12 ×
41/2/2/49 + 16 ×
41/2/3/39 + 18 ×
41/1/2/610 + 12 ×
41/1/3/510 + 15 ×
41/2/2/510 + 20 ×
41/2/3/410 + 24 ×
41/1/3/611 + 18 ×
41/1/4/511 + 20 ×
41/2/2/611 + 24 ×
41/2/3/511 + 30 ×
41/2/4/411 + 32 ×
41/3/3/411 + 36 ×
42/2/3/411 + 48 ×
41/1/4/612 + 24 ×
41/2/3/612 + 36 ×
41/2/4/512 + 40 ×
41/3/3/512 + 45 ×
41/3/4/412 + 48 ×
42/2/3/512 + 60 ×
42/3/3/412 + 72 ×
41/1/5/613 + 30 ×
41/2/4/613 + 48 ×
41/2/5/513 + 50 ×
41/3/3/613 + 54 ×
41/3/4/513 + 60 ×
42/2/3/613 + 72 ×
42/2/4/513 + 80 ×
42/3/3/513 + 90 ×
42/3/4/413 + 96 ×
41/2/5/614 + 60 ×
41/3/4/614 + 72 ×
41/3/5/514 + 75 ×
41/4/4/514 + 80 ×
42/2/4/614 + 96 ×
42/3/3/614 + 108 ×
42/3/4/514 + 120 ×
41/2/6/615 + 72 ×
41/3/5/615 + 90 ×
41/4/4/615 + 96 ×
41/4/5/515 + 100 ×
42/2/5/615 + 120 ×
42/3/4/615 + 144 ×
42/3/5/515 + 150 ×
42/4/4/515 + 160 ×
43/3/4/515 + 180 ×
41/3/6/616 + 108 ×
41/4/5/616 + 120 ×
42/3/5/616 + 180 ×
42/4/4/616 + 192 ×
42/4/5/516 + 200 ×
43/3/4/616 + 216 ×
43/4/4/516 + 240 ×
41/4/6/617 + 144 ×
41/5/5/617 + 150 ×
42/3/6/617 + 216 ×
42/4/5/617 + 240 ×
43/3/5/617 + 270 ×
43/4/4/617 + 288 ×
43/4/5/517 + 300 ×
41/5/6/618 + 180 ×
42/4/6/618 + 288 ×
42/5/5/618 + 300 ×
43/4/5/618 + 360 ×
42/5/6/619 + 360 ×
43/4/6/619 + 432 ×
43/5/5/619 + 450 ×
44/4/5/619 + 480 ×
43/5/6/620 + 540 ×
44/5/5/620 + 600 ×
44/5/6/621 + 720 ×Unique
51/1/2/3/411 + 24 ×Unique
51/1/2/3/512 + 30 ×
51/2/2/3/412 + 48 ×
51/1/2/3/613 + 36 ×
51/1/2/4/513 + 40 ×
51/2/2/3/513 + 60 ×
51/2/3/3/413 + 72 ×
51/1/2/4/614 + 48 ×
51/1/3/4/514 + 60 ×
51/2/2/3/614 + 72 ×
51/2/2/4/514 + 80 ×
51/2/3/3/514 + 90 ×
51/2/3/4/414 + 96 ×
51/1/2/5/615 + 60 ×
51/1/3/4/615 + 72 ×
51/2/2/4/615 + 96 ×
51/2/3/3/615 + 108 ×
51/2/3/4/515 + 120 ×
51/1/3/5/616 + 90 ×
51/2/2/5/616 + 120 ×
51/2/3/4/616 + 144 ×
51/2/3/5/516 + 150 ×
51/2/4/4/516 + 160 ×
51/3/3/4/516 + 180 ×
52/2/3/4/516 + 240 ×
51/1/4/5/617 + 120 ×
51/2/3/5/617 + 180 ×
51/2/4/4/617 + 192 ×
51/2/4/5/517 + 200 ×
51/3/3/4/617 + 216 ×
51/3/4/4/517 + 240 ×
52/2/3/4/617 + 288 ×
52/3/3/4/517 + 360 ×
51/2/3/6/618 + 216 ×
51/2/4/5/618 + 240 ×
51/3/3/5/618 + 270 ×
51/3/4/4/618 + 288 ×
51/3/4/5/518 + 300 ×
52/2/3/5/618 + 360 ×
52/3/3/4/618 + 432 ×
52/3/4/4/518 + 480 ×
51/2/4/6/619 + 288 ×
51/2/5/5/619 + 300 ×
51/3/4/5/619 + 360 ×
52/2/4/5/619 + 480 ×
52/3/3/5/619 + 540 ×
52/3/4/4/619 + 576 ×
52/3/4/5/519 + 600 ×
51/2/5/6/620 + 360 ×
51/3/4/6/620 + 432 ×
51/3/5/5/620 + 450 ×
51/4/4/5/620 + 480 ×
52/3/4/5/620 + 720 ×
51/3/5/6/621 + 540 ×
51/4/5/5/621 + 600 ×
52/3/4/6/621 + 864 ×
52/3/5/5/621 + 900 ×
52/4/4/5/621 + 960 ×
53/3/4/5/621 + 1080 ×
51/4/5/6/622 + 720 ×
52/3/5/6/622 + 1080 ×
52/4/5/5/622 + 1200 ×
53/4/4/5/622 + 1440 ×
52/4/5/6/623 + 1440 ×
53/4/5/5/623 + 1800 ×
53/4/5/6/624 + 2160 ×Unique

Two dog-leg 6x6 Cage Combos

Cage SizeSpread+×-÷
31/1/24 + 2 ×Unique
31/1/35 + 3 ×
31/2/25 + 4 ×
31/1/46 + 4 ×
31/2/36 + 6 ×
31/1/57 + 5 ×
31/2/47 + 8 ×
31/3/37 + 9 ×
32/2/37 + 12 ×
31/1/68 + 6 ×
31/2/58 + 10 ×
31/3/48 + 12 ×
32/2/48 + 16 ×
32/3/38 + 18 ×
31/2/69 + 12 ×
31/3/59 + 15 ×
31/4/49 + 16 ×
32/2/59 + 20 ×
32/3/49 + 24 ×
31/3/610 + 18 ×
31/4/510 + 20 ×
32/2/610 + 24 ×
32/3/510 + 30 ×
32/4/410 + 32 ×
33/3/410 + 36 ×
31/4/611 + 24 ×
31/5/511 + 25 ×
32/3/611 + 36 ×
32/4/511 + 40 ×
33/3/511 + 45 ×
33/4/411 + 48 ×
31/5/612 + 30 ×
32/4/612 + 48 ×
32/5/512 + 50 ×
33/3/612 + 54 ×
33/4/512 + 60 ×
31/6/613 + 36 ×
32/5/613 + 60 ×
33/4/613 + 72 ×
33/5/513 + 75 ×
34/4/513 + 80 ×
32/6/614 + 72 ×
33/5/614 + 90 ×
34/4/614 + 96 ×
34/5/514 + 100 ×
33/6/615 + 108 ×
34/5/615 + 120 ×
34/6/616 + 144 ×
35/5/616 + 150 ×
35/6/617 + 180 ×Unique
41/1/2/26 + 4 ×Unique
41/1/2/37 + 6 ×Unique
41/1/2/48 + 8 ×
41/1/3/38 + 9 ×
41/2/2/38 + 12 ×
41/1/2/59 + 10 ×
41/1/3/49 + 12 ×
41/2/2/49 + 16 ×
41/2/3/39 + 18 ×
41/1/2/610 + 12 ×
41/1/3/510 + 15 ×
41/1/4/410 + 16 ×
41/2/2/510 + 20 ×
41/2/3/410 + 24 ×
42/2/3/310 + 36 ×
41/1/3/611 + 18 ×
41/1/4/511 + 20 ×
41/2/2/611 + 24 ×
41/2/3/511 + 30 ×
41/2/4/411 + 32 ×
41/3/3/411 + 36 ×
42/2/3/411 + 48 ×
41/1/4/612 + 24 ×
41/1/5/512 + 25 ×
41/2/3/612 + 36 ×
41/2/4/512 + 40 ×
41/3/3/512 + 45 ×
41/3/4/412 + 48 ×
42/2/3/512 + 60 ×
42/2/4/412 + 64 ×
42/3/3/412 + 72 ×
41/1/5/613 + 30 ×
41/2/4/613 + 48 ×
41/2/5/513 + 50 ×
41/3/3/613 + 54 ×
41/3/4/513 + 60 ×
42/2/3/613 + 72 ×
42/2/4/513 + 80 ×
42/3/3/513 + 90 ×
42/3/4/413 + 96 ×
41/1/6/614 + 36 ×
41/2/5/614 + 60 ×
41/3/4/614 + 72 ×
41/3/5/514 + 75 ×
41/4/4/514 + 80 ×
42/2/4/614 + 96 ×
42/2/5/514 + 100 ×
42/3/3/614 + 108 ×
42/3/4/514 + 120 ×
43/3/4/414 + 144 ×
41/2/6/615 + 72 ×
41/3/5/615 + 90 ×
41/4/4/615 + 96 ×
41/4/5/515 + 100 ×
42/2/5/615 + 120 ×
42/3/4/615 + 144 ×
42/3/5/515 + 150 ×
42/4/4/515 + 160 ×
43/3/4/515 + 180 ×
41/3/6/616 + 108 ×
41/4/5/616 + 120 ×
42/2/6/616 + 144 ×
42/3/5/616 + 180 ×
42/4/4/616 + 192 ×
42/4/5/516 + 200 ×
43/3/4/616 + 216 ×
43/3/5/516 + 225 ×
43/4/4/516 + 240 ×
41/4/6/617 + 144 ×
41/5/5/617 + 150 ×
42/3/6/617 + 216 ×
42/4/5/617 + 240 ×
43/3/5/617 + 270 ×
43/4/4/617 + 288 ×
43/4/5/517 + 300 ×
41/5/6/618 + 180 ×
42/4/6/618 + 288 ×
42/5/5/618 + 300 ×
43/3/6/618 + 324 ×
43/4/5/618 + 360 ×
44/4/5/518 + 400 ×
42/5/6/619 + 360 ×
43/4/6/619 + 432 ×
43/5/5/619 + 450 ×
44/4/5/619 + 480 ×
43/5/6/620 + 540 ×
44/4/6/620 + 576 ×
44/5/5/620 + 600 ×
44/5/6/621 + 720 ×Unique
45/5/6/622 + 900 ×Unique
51/1/2/2/39 + 12 ×Unique
51/1/2/2/410 + 16 ×
51/1/2/3/310 + 18 ×
51/2/2/2/310 + 24 ×
51/1/2/2/511 + 20 ×
51/1/2/3/411 + 24 ×
51/2/2/2/411 + 32 ×
51/2/2/3/311 + 36 ×
51/1/2/2/612 + 24 ×
51/1/2/3/512 + 30 ×
51/1/2/4/412 + 32 ×
51/1/3/3/412 + 36 ×
51/2/2/2/512 + 40 ×
51/2/2/3/412 + 48 ×
51/2/3/3/312 + 54 ×
51/1/2/3/613 + 36 ×
51/1/2/4/513 + 40 ×
51/1/3/3/513 + 45 ×
51/1/3/4/413 + 48 ×
51/2/2/2/613 + 48 ×
51/2/2/3/513 + 60 ×
51/2/2/4/413 + 64 ×
51/2/3/3/413 + 72 ×
51/1/2/4/614 + 48 ×
51/1/2/5/514 + 50 ×
51/1/3/3/614 + 54 ×
51/1/3/4/514 + 60 ×
51/2/2/3/614 + 72 ×
51/2/2/4/514 + 80 ×
51/2/3/3/514 + 90 ×
51/2/3/4/414 + 96 ×
51/3/3/3/414 + 108 ×
52/2/3/3/414 + 144 ×
51/1/2/5/615 + 60 ×
51/1/3/4/615 + 72 ×
51/1/3/5/515 + 75 ×
51/1/4/4/515 + 80 ×
51/2/2/4/615 + 96 ×
51/2/2/5/515 + 100 ×
51/2/3/3/615 + 108 ×
51/2/3/4/515 + 120 ×
51/2/4/4/415 + 128 ×
51/3/3/3/515 + 135 ×
51/3/3/4/415 + 144 ×
52/2/3/3/515 + 180 ×
52/2/3/4/415 + 192 ×
52/3/3/3/415 + 216 ×
51/1/2/6/616 + 72 ×
51/1/3/5/616 + 90 ×
51/1/4/4/616 + 96 ×
51/1/4/5/516 + 100 ×
51/2/2/5/616 + 120 ×
51/2/3/4/616 + 144 ×
51/2/3/5/516 + 150 ×
51/2/4/4/516 + 160 ×
51/3/3/3/616 + 162 ×
51/3/3/4/516 + 180 ×
51/3/4/4/416 + 192 ×
52/2/3/3/616 + 216 ×
52/2/3/4/516 + 240 ×
52/3/3/3/516 + 270 ×
52/3/3/4/416 + 288 ×
51/1/3/6/617 + 108 ×
51/1/4/5/617 + 120 ×
51/2/2/6/617 + 144 ×
51/2/3/5/617 + 180 ×
51/2/4/4/617 + 192 ×
51/2/4/5/517 + 200 ×
51/3/3/4/617 + 216 ×
51/3/3/5/517 + 225 ×
51/3/4/4/517 + 240 ×
52/2/3/4/617 + 288 ×
52/2/3/5/517 + 300 ×
52/2/4/4/517 + 320 ×
52/3/3/3/617 + 324 ×
52/3/3/4/517 + 360 ×
52/3/4/4/417 + 384 ×
51/1/4/6/618 + 144 ×
51/1/5/5/618 + 150 ×
51/2/3/6/618 + 216 ×
51/2/4/5/618 + 240 ×
51/2/5/5/518 + 250 ×
51/3/3/5/618 + 270 ×
51/3/4/4/618 + 288 ×
51/3/4/5/518 + 300 ×
51/4/4/4/518 + 320 ×
52/2/3/5/618 + 360 ×
52/2/4/4/618 + 384 ×
52/2/4/5/518 + 400 ×
52/3/3/4/618 + 432 ×
52/3/3/5/518 + 450 ×
52/3/4/4/518 + 480 ×
51/1/5/6/619 + 180 ×
51/2/4/6/619 + 288 ×
51/2/5/5/619 + 300 ×
51/3/3/6/619 + 324 ×
51/3/4/5/619 + 360 ×
51/3/5/5/519 + 375 ×
51/4/4/4/619 + 384 ×
51/4/4/5/519 + 400 ×
52/2/3/6/619 + 432 ×
52/2/4/5/619 + 480 ×
52/3/3/5/619 + 540 ×
52/3/4/4/619 + 576 ×
52/3/4/5/519 + 600 ×
52/4/4/4/519 + 640 ×
53/3/4/4/519 + 720 ×
51/2/5/6/620 + 360 ×
51/3/4/6/620 + 432 ×
51/3/5/5/620 + 450 ×
51/4/4/5/620 + 480 ×
51/4/5/5/520 + 500 ×
52/2/4/6/620 + 576 ×
52/2/5/5/620 + 600 ×
52/3/3/6/620 + 648 ×
52/3/4/5/620 + 720 ×
52/3/5/5/520 + 750 ×
52/4/4/4/620 + 768 ×
52/4/4/5/520 + 800 ×
53/3/4/4/620 + 864 ×
53/3/4/5/520 + 900 ×
53/4/4/4/520 + 960 ×
51/2/6/6/621 + 432 ×
51/3/5/6/621 + 540 ×
51/4/4/6/621 + 576 ×
51/4/5/5/621 + 600 ×
52/2/5/6/621 + 720 ×
52/3/4/6/621 + 864 ×
52/3/5/5/621 + 900 ×
52/4/4/5/621 + 960 ×
52/4/5/5/521 + 1000 ×
53/3/4/5/621 + 1080 ×
53/4/4/4/621 + 1152 ×
53/4/4/5/521 + 1200 ×
51/3/6/6/622 + 648 ×
51/4/5/6/622 + 720 ×
51/5/5/5/622 + 750 ×
52/3/5/6/622 + 1080 ×
52/4/4/6/622 + 1152 ×
52/4/5/5/622 + 1200 ×
53/3/4/6/622 + 1296 ×
53/3/5/5/622 + 1350 ×
53/4/4/5/622 + 1440 ×
53/4/5/5/522 + 1500 ×
51/4/6/6/623 + 864 ×
51/5/5/6/623 + 900 ×
52/3/6/6/623 + 1296 ×
52/4/5/6/623 + 1440 ×
52/5/5/5/623 + 1500 ×
53/3/5/6/623 + 1620 ×
53/4/4/6/623 + 1728 ×
53/4/5/5/623 + 1800 ×
51/5/6/6/624 + 1080 ×
52/4/6/6/624 + 1728 ×
52/5/5/6/624 + 1800 ×
53/4/5/6/624 + 2160 ×
53/5/5/5/624 + 2250 ×
54/4/5/5/624 + 2400 ×
52/5/6/6/625 + 2160 ×
53/4/6/6/625 + 2592 ×
53/5/5/6/625 + 2700 ×
54/4/5/6/625 + 2880 ×
54/5/5/5/625 + 3000 ×
53/5/6/6/626 + 3240 ×
54/5/5/6/626 + 3600 ×
54/5/6/6/627 + 4320 ×Unique

Go back to Killer CombinationsContinue to KenKen Dog Legs


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CommentsTalk

... by: Ken

Tuesday 17-Feb-2015
I've been learning KenKen and am pretty new to it. One 5x5 has me stumped. It is supposed to be "easy." I can solve it only by assuming one of the solutions in a 13+ dogleg cage with three squares: 454 works but 535 does not. Trying your various strategies to eliminate the 3 or 5 and thus the 535 dogleg seems the best approach. I have not been able to see one though. I see you've restricted yourself to 6x6 KKs. However I would appreciate it if I could email it to you and get your take. Your efforts in Sudoku and KK are astounding!
Thanks, Ken Rogers
Andrew Stuart writes:
I'd be very happy to take a look. Use the "Email This Board" button and copy paste to me.
Add to this Thread

... by: yves

Friday 4-Mar-2011
Please advice me when this list of combination is available.
I imagine, it will be possible to print the whole list or download a PDF.

Thanks in advance
Andrew Stuart writes:
Ok documentation page completed! Emailed you
Add to this Thread
Article created on 3-July-2009. Views: 43715
This page was last modified on 1-September-2025.
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