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• Numbers are distinct; their sum is 100. Every cell is part of a numbered region.
• (guessing) It seems that 28 must be unshaded and the other numbers shaded, or there will be some sort of violation (can't prove this part).

enter image description here

• Numbers are distinct; their sum is 100. Every cell is part of a numbered region.
• (guessing) It seems that 28 must be unshaded and the other numbers shaded, or there will be some sort of violation (can't prove this part).
• The 24 can be 3x8 or 4x6. In any case, R1C8-R3C10 are shaded.
• The 12 can be 2x6 or 3x4. In any case, R1C1-R2C2 are shaded.
• The 20 can be 2x10 or 4x5. In any case, R9C9-R10C10 are shaded.
• If 24 is a 3x8, it will border the 12 or 20; it must be 4x6; R1C7-R4C10 are shaded.
• If the 12 is a 2x6, is will border the 16 or 24; it is 3x4; R1C1-R3C3 are shaded.
• If the 24 is R1C7-R6C10, it forces the 20 to be R9C1-R10C10, and the 16 can not be completed; the 24 is R1C5-R4C10.
• The 12 is forced to be R1C1-R4C3.
• R5C4 is shaded, or the 28 isn't big enough.

On puzzle 1, I count 40 ambiguous cells, and 9 solutions (3 with the 20 at R9C1-R10C10, 2 with the 20 at R7C6-R10C10, 4 with the 20 at R6C7-R10C10).
On puzzle 2, I count 2 solutions with 2 ambiguous cells.
On puzzle 3, 2 solutions with 14 ambiguous cells.
On puzzle 4, 3 solutions with 3 ambiguous cells.
On puzzle 5, 4 solutions with 8 ambiguous cells.
On puzzle 6, 2 solutions with 2 ambiguous cells. (and Sny's signature)
On puzzle 7, 18 solutions with 18 ambiguous.
On puzzle 8, 2 solutions with 2 ambiguous.
On puzzle 9, 4 solutions with 10 ambiguous.

Puzzle #2:

Basic deductions:

Following those directions leads to (cb-2a.png)

(cb-2g.png)

Basic deductions:

Puzzle #1:

• Numbers are distinct; their sum is 100. Every cell is part of a numbered region.
• (guessing) It seems that 28 must be unshaded and the other numbers shaded, or there will be some sort of violation (can't prove this part).

Puzzle #2:

Following those basic deductions leads to (cb-2a.png)

(cb-2g.png)

#3:

• The 1s and 2s are shaded; some unshaded cells are forced.
• However the 2 is positioned, R3C3 is unshaded.
• If R3C2 were unshaded, that would force R4C1-R4C3 shaded, but that couldn't be expanded to a rectangle of 4; R3C2 is shaded.
• Completion of 2 and 4s is forced.
• The 8 is unshaded, or it is forced to a nonrectangular shape.
• R4C5 is shaded (adjacent to unshaded)
• R5C3 is unshaded (or would make nonrectangular shaded area)
• R1C5-R3C5 are unshaded (borders a shaded area)
• R5C4 is unshaded (only way to extend the 8)
• R6C3 is shaded (borders the 8)
• R6C1, R6C2 unshaded (borders completed region)
• However R4C5 and R6C3 go, R5C6 and R7C4 are unshaded.
• However the 8 is extended, it will border R6C5, so that cell is shaded.

• If R6C5 is not connected to either 3, then its borders force too many unshaded cells to be part of the 8; R6C6 and R7C5 are unshaded.
• Whichever way it does get connected quickly forces the other 3, the rest of the 8, and two 1x2 regions. After that, there are exactly 65 unclaimed cells left, just enough.

enter image description here

#9:

• The distinct numbers sum to 99. The 32s are in the same region, and there is a single un-numbered region (1 cell, shaded).
• The 32s must be unshaded (no rectangle can hold both)
• Every other numbered region is shaded (either they border the 32, or it would take too many shaded cells between it and the 32).
• There's only one way to do the 7.
• The 36 can only be done as a 4x9 rectangle, and there are only two ways to place it, which force some cells.
• If R1C5 or R10C5 is shaded, that forces an unshaded rectangle in the upper left or lower left corner.
• The 6 is forced.