Detailed solutions

(The first several deductions are the same for both Nurikabe #8 and #9.)

In the following,

• 'a' denotes a cell which must be white to allow white chain to grow.
• 'B' denotes a cell which must be black to allow black chain to connect.
• 'C' denotes a cell which must be black to provide border for white chain.
• 'D' denotes a cell which must be black to separate two white chains.
• 'e' denotes a cell which must be white to prevent 2-x-2 black square.
• 'F' denotes a cell which cannot be reached by any white chain.

A black group in lower left is almost isolated, so linking black cells can be deduced across the next-to-last row.

				1		a			a		B
		2				6	a			1
			3				C	5
		4				1						7
1		e		2
			1					3		1
	4										1
		3		C			5			1
a		e		B							1
C			8		a
		2		B	D	B
			B		7/8	a	a					2

The next diagram shows our first non-trivial deduction. The point y must be white: Otherwise only the 5-chain could prevent a 2-x-2 black square starting above the y and in doing so would be unable to prevent 2-x-2 black square near the z.

The point z must also be white to prevent 2-x-2 black square.

				1
		2				6				1
F			3					5	B		B
C		4				1		a				7
	C	a	C
1				2	y
			1					3		1
	4				z						1
		3					5			1
											1
			8
		2
				e	7/8							2

 
 

				1
		2				6				1
			3					5
		4		e		1						7
				C	C
1				2		C
			1		C			3		1
	4										1
		3					5			1
											1
			8
		2
					7/8							2

The next diagram demonstrates that the black chain in lower left has no connections to blacks in upper part of board except via a route below the 5-chain. If such a connection route existed it would use the cells shown in green with G, and the 8-chain would not be able to grow to full size without isolating three black cells to its left.

				1
		2				6				1
			3		c			5
		4				1						7
1				2
			1					3		1
	4					G					1
		3			a	G	5	a		1
					a	G	G	a			1
			8			a	G
		2					G			e
					7/8			a				2

To break up the 2-x-2 black square at G-x-y-x in lower right, either y or G must be white. These two possibilities lead to our two solutions but first we make deductions common to both cases.

If G is white, the black chain at bottom has no outlet except to the right and the cells B at center right must be black for connection. If y is white, the black chain connects above it (shown in green), e is white (to prevent a 2-x-2 black square) and again the only outlet for the black chain is with the cells B at center right.

Meanwhile, the black chain in upper right must connect via M-M, N-N or P-P. M-M is out because 5-chain wouldn't be able to grow to full size; and we can now see that P-P is out because 7-chain wouldn't be able to grow to full size. So N-N are black.

				1
		2				6				1
			3					5
		4				1			M	N	P	7
									M	N	P
1				2							B	e
			1					3		1		B
	4										1
		3					5			1
									e		1
			8							G	x
		2								y	x
					7/8							2

 
 

				1								C
		2				6				1
			3					5
		4				1			e		e	7
								a			e	a
1				2				C	B
			1				a	3		1
	4						D				1
		3					5			1
											1
			8							A
		2								B
					7/8							2

The above diagram brings us up to date. Recall that one of the reddish cells (A or B) in lower right is white. We pursue these cases separately. Note that A can connect only to the upper 8-chain, while B can connect only to the lower 8-chain.

Case A

Assuming A is white we quickly derive solution as shown. Note that the bottom chain must be a 7, not an 8.

				1
		2				6				1
			3					5
		4				1						7
									e
1				2			e
			1					3		1
	4							C			1
		3			e	a	5	e		1
					C	C	C	C	C		1
			8							A
		2					C	C	C	C
					7			a	a	a		2

Case B

Assuming B is white we recall deductions mentioned above. Note that the bottom chain must be an 8, not a 7.

The white cell 'x' must be part of the 8-chain, so the cell to its left must be black for connection.

				1
		2				6				1
			3					5
		4				1						7
									e
1				2			e
			1					3		1
	4			B	x						1
		3					5			1
								a	e		1
			8						B	B
		2								B
					8							2

In the following note that 'G' (in green) must be black or the two large black groups are unconnected.
 
 

				1
		2				6				1
			3					5
		4				1						7
							C
1				2
			1			C		3		1
	4					G		C			1
		3			a	D	5	e		1
					a						1
			8
		2
					8							2

 
 

				1
		2				6				1
			3					5
		4				1						7
1				2
			1					3		1
	4										1
		3					5			1
							D				1
			8			a	D	a
		2					B	D	B
					8			a	a			2

 
 
 
 

These puzzles are Copyright © 2007 by James Dow Allen