here is my contribution

are you asking for minimum number of cages, or minimum number of cells occupied by cages?

it turns out that you can use the approach for minimum givens on larger sudokus in

this thread to create killer sudokus. the trick is to notice that replacing each "row" of givens by one cage with the biggest or smallest possible sums forces all the cages to be solved one after another

if you allow for repeated digits in cages, for the 3x3 case you get a 1-cage killer sudoku that looks like this

- Code: Select all
`X........`

XXXX.....

XXXXXXX..

XX.......

XXXXX....

XXXXXXXX.

XXX......

XXXXXX...

XXXXXXXXX

X = 165

the solution is isomorphic to the canonical grid. i actually came up with this killer sudoku myself in 2012 and posted it to my facebook, but it only got 8 likes and 12 comments (5 of the comments were me) so i'm sorely disappointed that my facebook friends aren't as enthused by low clue killers as you guys are. i didn't post it here because this forum was completely dead in 2012. i also cross posted it to motris's blog

herenow if you disallow repeats in cages, you can split this cage into separate cages in each row, and get rid of the last cage because it doesn't do anything. then if you want to consider least cells in cages you can move the cages with 5-8 cells to the other side of the row to make them smaller. so you get this (i've shuffled the rows and columns to make it symmetric)

- Code: Select all
`AA.......`

........B

.....CCCC

DDD......

.........

......EEE

FFFF.....

G........

.......HH

A = 17

B = 1

C = 10

D = 24

E = 6

F = 30

G = 9

H = 3

which has 8 cages, 20 occupied cells, and is trivially easy to solve

by tarek's standards this thing has a tiny flaw in that there are two one-cell cages, so i'm thinking of the most efficient way to get rid of them. the "obvious" way to do it is to replace the one-cell cages by their row complements to get this 8-cage killer with no singles:

- Code: Select all
`AA.......`

BBBBBBBB.

.....CCCC

DDD......

.........

......EEE

FFFF.....

.GGGGGGGG

.......HH

A = 17

B = 44

C = 10

D = 24

E = 6

F = 30

G = 36

H = 3

however, if you care about the minimum number of occupied cells, this wouldn't be optimal. i've come up with a way to replace the single cages with two doubles each, if you allow diagonal cages like hatman is doing. the solution of this one is the same as the one above except i swapped c1 with c2 and c8 with c9 to get the diagonal cages to line up

- Code: Select all
`AAII..B..`

.......B.

.....CCCC

DDD......

.........

......EEE

FFFF.....

.G.......

..G..JJHH

A = 17

B = 6

C = 10

D = 24

E = 6

F = 30

G = 14

H = 3

I = 3

J = 17

which has 26 occupied cells and no singles. i'm wondering whether a better design is still possible for this idea with the canonical grid. unfortunately any of these puzzles will be trivially easy and boring to solve

hatman, your link doesn't work

edit: fixed typos and explicitly added the 8-cage no singles killer