The easiest place to start is with the black squares with the "4" in them. It is helpful to draw a line through any illuminated square from the light bulb and an "X" through the ones that cannot contain a light bulb.
FILLOMINO~
Whenever there are two 2's diagonal to eachother and you know by the rules that they cannot horizontally or vertically share sides, they must be seperated from eachother.
The most basic first move for filling in a nonogram puzzle is to mark the rows/column the closest to one side of the grid and coming from the other side of the grid filling in the whole row as tight knit as you can. Then the parts of the SAME run that overlap from one side to the other must be filled in.
HEX TAKEGAKI~
Note: The following pictures are used as a visual guide and are not full puzzles.
Figure A: We know that there are no acute angles in the loop, only obtuse angles are used.
Figure B: I always find it helpful to shade in the areas that I know will not be a part of the loop.
Figure C: Any time there is a "1" on a side of a puzzle you know that the hexagons beside it on the side are not a part of the loop.
Figure D. Anytime there is a "3" in a corner of the puzzle it looks like this.
Figure E. If there is a "4", "5", or "6" in the hexagon directly adjacent to the corner, you know that the loop must go through the three hexagons around the corner.
In Kakuro, some of the totals only have one unique solution for the number of spaces given for the answer. For instance the clue 3 can ONLY be summed up by adding a 1 and 2. Here are a list of these unique solutions and their corresponding number of spaces for the answer. The Kakuro solution can have these digits in any order for the answer.
CLUE = ANSWER (# OF SPACES NEEDED FOR UNIQUE ANSWER)
3 = 1, 2 (2)
4 = 1, 3 (2)
6 = 1, 2, 3 (3)
7 = 1, 2, 4 (3)
10 = 1, 2, 3, 4 (4)
11 = 1, 2, 3, 5 (4)
15 = 1, 2, 3, 4, 5 (5)
16 = 9, 7 (2)
16 = 1, 2, 3, 4, 6 (5)
17 = 9, 8 (2)
21 = 1, 2, 3, 4, 5, 6 (6)
22 = 1, 2, 3, 4, 5, 7 (6)
23 = 9, 8, 6 (3)
24 = 9, 8, 7 (3)
28 = 1, 2, 3, 4, 5, 6, 7 (7)
29 = 1, 2, 3, 4, 5, 6, 8 (7)
29 = 9, 8, 7, 5 (4)
30 = 9, 8, 7, 6 (4)
34 = 9, 8, 7, 6, 4 (5)
35 = 9, 8, 7, 6, 5 (5)
36 = 1, 2, 3, 4, 5, 6, 7, 8 (8)
37 = 1, 2, 3, 4, 5, 6, 7, 9 (8)
38 = 9, 8, 7, 6, 5, 3 (6)
38 = 1, 2, 3, 4, 5, 6, 8, 9 (8)
39 = 9, 8, 7, 6, 5, 4 (6)
39 = 1, 2, 3, 4, 5, 7, 8, 9 (8)
40 = 1, 2, 3, 4, 6, 7, 8, 9 (8)
41 = 9, 8, 7, 6, 5, 4, 2 (7)
41 = 1, 2, 3, 5, 6, 7, 8, 9 (8)
42 = 9, 8, 7, 6, 5, 4, 3 (7)
42 = 1, 2, 4, 5, 6, 7, 8, 9 (8)
43 = 9, 8, 7, 6, 5, 4, 3, 1 (8)
44 = 9, 8, 7, 6, 5, 4, 3, 2 (8)
45 = 1, 2, 3, 4, 5, 6, 7, 8, 9 (9)
SHIKAKU ~
All prime numbers are contained in a 1x rectangle. Prime numbers are 2, 3, 5, 7, 11, 13, 17 etc.
Odd numbers cannot be in a 2x rectangle (Example: 9 = 1x9 or 3x3).
Anytime you have a number beside another number, you know that they are contained in different areas, therefore you know that a line goes between them. This is shown with the 3 and 4 in the sample puzzle below.
TAKEGAKI ~
Whenever there are two 3's side by side in the grid we can use a method of contradiction to know that there must be lines of the loop between the 3's and on the outside of each 3...
FIGURE B - I try two scenarios here where the segment that is not contained in the loop (yellow lines) are on the outside of each three. This scenario contradicts itself because we know that the loop cannot touch itself (red lines are contradiction lines).
FIGURE C - I try the open segment (yellow) in between the two 3's and automatically come up with a contradiction because the loop closed itself in around the two 3's.
FIGURE D - I now try the only remaining two scenarios... and they work!! So now we know that anytime two or more 3's are side by side in a grid, it will look like this snake pattern.
FIGURE E - Because we had proven that the open space CANNOT be on the outsides of the 3's (figure B) and CANNOT be in between the 3's (figure C), we then KNOW that these segments must be lines in the loop.
