Introduction
Killer Sudoku (also known as killer su doku, sumdoku, sum doku, addoko, or samunamupure) is derived from sudoku and kakuro. First developed in Japan in the mid 1990s as “samunamupure” (translated to 'sum number place'), the puzzle has been 'labeled' by the London Times as a 'killer' puzzle due to its complexity & frustrating complexity.

Fig. 1  Killer Sudoku Puzzle

Terminology
- cell -- a single square that contains one number in the puzzle
- row (r) -- a horizontal line of 9 cells
- column (c) -- a vertical line of 9 cells
- nonet (N) -- a 3x3 grid of cells, outlined by borders (also referred to as a 'box' in Sudoku puzzles). The term nonet eliminates potential ambiguity between the words “box” and “square”
- cage -- a group of internal cells outlined by a dotted line (or by individual colours)
    - Abbreviated cell description: To describe a cage with a sum of 23 that spans 3 cells, one could use a 'longhand' description of “23-sum 3-cell cage”, but for brevity I'll describe this as “23(3)”, and the set of those 3 numbers can be as an ordered set: [689] (i.e. in square brackets - required in exactly that order), or as {689} (with curly brackets, to denote an unordered set in any order)
- region (or house) -- any non-repeating set of 9 cells (can be used as a general term for 'row, cell or nonet', and in Diagonal Killer (similar to X-Factor Sudoku) as 'long diagonal')

Rules
The objective is to fill the puzzle grid with numbers from 1 to 9 so that the following conditions are met:
- Each row, column and nonet contains each number exactly once;
- The sum of all numbers in a cage must match the small number printed in the cage's upper left corner;
- No number is repeated in a cage (i.e. no cage can contain more than 9 cells but may overlap nonets);
- In Diagonal Killer Sudoku (also known as 'Killer X'), each of the long diagonals contains each number once;

- Rule of 45: Each Sudoku region contains the digits one through nine, adding up to 45. If X is the sum of all the cages contained entirely in a region, then the cells not covered must sum to 45-X. By adding up the cages and single numbers in a particular region, one can deduce the result of a single cell. If the cell calculated is within the region itself, it's referred to as an 'innie' (Fig. 1 - the top-left nonet has 1 innie from 9(2)); if outside the region, it's called an 'outie' (Fig. 1 - top-right nonet has outties from 36(6)). The 'rule of 45' can be extended to calculate the innies or outies of N adjacent regions - as the difference between the cage sums and N*45. Often it's useful to derive the sum of 2 or 3 cells, then use other elimination techniques.

Hints
- Since Killer Puzzles are derived from regular Sudoku Puzzles, don't totally rely on Killer Sudoku rules alone to solve a Killer Sudoku puzzle - use common Sudoku solution techniques as well (i.e. Naked/Hidden Pairs, X-Wing, Colouring, Swordfish, Forcing Chains etc.) may well help you to solve the most difficult Killer Sudoku puzzle;

- Look for the Fewest Possible Combinations

3	12	3
4	13	4
5	14	23	5
6	15	24	123	6
7	16	25	34	124	7
8	17	26	35	125	134	8
9	18	27	36	45	126	135	234	9
10	19	28	37	46	127	136	145	235	1234	10
11	29	38	47	56	128	137	146	236	245	1235	11
12	39	48	57	129	138	147	156	237	246	345	1236	1245	12
13	49	58	67	139	148	157	238	247	256	346	1237	1246	1345	13
14	59	68	149	158	167	239	248	257	347	356	1238	1247	1256	1346	2345	14
15	69	78	159	168	249	258	267	348	357	456	1239	1248	1257	1347	1356	2346	12345	15
16	79	169	178	259	268	349	358	367	457	1249	1258	1267	1348	1357	1456	2347	2356	12346	16
17	89	179	269	278	359	368	458	467	1259	1268	1349	1358	1367	1457	2348	2357	2456	12347	12356	17
18	189	279	369	378	459	468	567	1269	1278	1359	1368	1458	1467	2349	2358	2367	2457	3456	12348	12357	12456	18
19	289	379	469	478	568	1279	1369	1378	1459	1468	1567	2359	2368	2458	2467	3457	12349	12358	12367	12457	13456	19
20	389	479	569	578	1289	1379	1469	1478	1568	2369	2378	2459	2468	2567	3458	3467	12359	12368	12458	12467	13457	23456	20
21	489	579	678	1389	1479	1569	1578	2379	2469	2478	2568	3459	3468	3567	12369	12378	12459	12468	12567	13458	13467	23457	123456	21
22	589	679	1489	1579	1678	2389	2479	2569	2578	3469	3478	3568	4567	12379	12469	12478	12568	13459	13468	13567	23458	23467	123457	22
23	689	1589	1679	2489	2579	2678	3479	3569	3578	4568	12389	12479	12569	12578	13469	13478	13568	14567	23459	23468	23567	123458	123467	23
24	789	1689	2589	2679	3489	3579	3678	4569	4578	12489	12579	12678	13479	13569	13578	14568	23469	23478	23568	24567	123459	123468	123567	24
	25	1789	2689	3589	3679	4579	4678	12589	12679	13489	13579	13678	14569	14578	23479	23569	23578	24568	34567	123469	123478	123568	124567	25
		26	2789	3689	4589	4679	5678	12689	13589	13679	14579	14678	23489	23579	23678	24569	24578	34568	123479	123569	123578	124568	134567	26
		27	3789	4689	5679	12789	13689	14589	14679	15678	23589	23679	24579	24678	34569	34578	123489	123579	123678	124569	124578	134568	234567	27
				28	4789	5689	13789	14689	15679	23689	24589	24679	25678	34579	34678	123589	123679	124579	124678	134569	134578	234568	1234567	28
					29	5789	14789	15689	23789	24689	25679	34589	34679	35678	123689	124589	124679	125678	134579	134678	234569	234578	1234568	29
						30	6789	15789	24789	25689	34689	35679	45678	123789	124689	125679	134589	134679	135678	234579	234678	1234569	1234578	30
								31	16789	25789	34789	35689	45679	124789	125689	134689	135679	145678	234589	234679	235678	1234579	1234678	31
										32	26789	35789	45689	125789	134789	135689	145679	234689	235679	245678	1234589	1234679	1235678	32
											33	36789	45789	126789	135789	145689	234789	235689	245679	345678	1234689	1235679	1245678	33
													34	46789	136789	145789	235789	245689	345679	1234789	1235689	1245679	1345678	34
														35	56789	146789	236789	245789	345689	1235789	1245689	1345679	2345678	35
																36	156789	246789	345789	1236789	1245789	1345689	2345679	36
																		37	256789	346789	1246789	1345789	2345689	37
																			38	356789	1256789	1346789	2345789	38
																				39	456789	1356789	2346789	39
																					40	1456789	2356789	40
																						41	2456789	41
																						42	3456789	42

Fig. 2  Killer Combination Table