Challenging Puzzle #51 - One Possible Solution Approach…
	NB:  There is only one final solution to this Sudoku puzzle…
	A	B	C	D	E	F	G	H	I
1
2	AA			BB			CC				Reference to 3x3 Blocks AA, BB & CC
3
4
5	DD			EE			FF				Reference to 3x3 Blocks DD, EE & FF
6
7
8	GG			HH			II				Reference to 3x3 Blocks GG, HH & II
9
	A	B	C	D	E	F	G	H	I
1						2					Challenging Puzzle #51 - November 20, 2005
2			6	9		5	3				Original puzzle as published © joe-ks.com
3		9	3	7			2	5
4		5	4		2			9
5			9		4		1	7
6			8		7			3
7		1	5				9	2
8		6	2	8		1	7
9				2
	A	B	C	D	E	F	G	H	I
1	5					2					Step 1: #5 is in Rows 2 & 3: #5 MUST go in Cell A1
2			6	9		5	3				(#5 in Column B @ B4 excludes use of Cell B1, &
3		9	3	7			2	5			#5 in Column C @ C7 excludes use of Cell C1)
4	7	5	4		2			9
5			9		4		1	7			Step 2: #7 is in Rows 5 & 6: #7 MUST go in Cell A4...
6			8		7			3
7		1	5				9	2
8		6	2	8		1	7
9				2
	A	B	C	D	E	F	G	H	I
1	5	478	17	1346	1368	2	468	1468	146789		Step 3: Determine all possible # combinations for vacant cells:
2	1248	2478	6	9	18	5	3	148	1478
3	148	9	3	7	168	468	2	5	1468		Obvious observations:  #2 in B6;  #4 in H8 ,& #7 in C9
4	7	5	4	136	2	368	68	9	68
5	236	23	9	356	4	368	1	7	2568		Step 4: re Block CC: #9 only shows up in Cell I1…
6	126	2	8	156	7	69	456	3	2456		Step 5: re Block DD: #1 only shows up in Cell A6…
7	348	1	5	346	36	3467	9	2	3468		Step 6: re Block EE: #9 only shows up in Cell F6...
8	349	6	2	8	359	1	7	4	345
9	3489	3478	7	2	3569	34679	4568	1468	134568
	A	B	C	D	E	F	G	H	I
1	5					2			9		Step 7: #2 is in Columns B & C: #2 MUST go in Cell A2
2	2		6	9		5	3				(#2 in Row 3 @ G3 excludes use of Cell A3)
3		9	3	7			2	5
4	7	5	4		2			9			Step 8: #7 is in Columns D & E: #7 MUST go in Cell F7
5			9		4		1	7	2		(#7 in Row 9 @ C9 excludes use of Cell F9)
6	1	2	8		7	9		3
7		1	5			7	9	2			Step 9: #2 is in Column G & H: #2 MUST go in Cell I5
8		6	2	8		1	7	4			(#2 in Row 4 @ E4 excludes use of Cell I5, &
9			7	2							#2 in Row 6 @ B6 excluides use of Cell I6)
	A	B	C	D	E	F	G	H	I
1	5		1			2			9		Step 10: Complete Column C: #1 MUST go in Cell C1...
2	2		6	9		5	3		7
3		9	3	7			2	5			Step 11: #7 is in Columns G & H: #7 MUST go in Cell I2
4	7	5	4		2			9			(#7 in Row 3 @ D3 excludes use of Cell I3)
5			9		4		1	7	2
6	1	2	8		7	9		3
7		1	5			7	9	2
8		6	2	8		1	7	4
9			7	2
	A	B	C	D	E	F	G	H	I
1	5	7	1			2			9		Step 12: Complete Block DD: need #s 3 & 6…
2	2		6	9		5	3		7		#6 can't go in Cell B5 (cf #6 in Column B @ B8), so
3		9	3	7			2	5			#6 MUST go in Cell A5, & therefore #3 MUST go in Cell B5
4	7	5	4		2			9
5	6	3	9		4		1	7	2		Step 13: #7 is in Rows 2 & 3: #7 MUST go in Cell B1...
6	1	2	8		7	9		3
7		1	5			7	9	2
8		6	2	8		1	7	4
9			7	2
	A	B	C	D	E	F	G	H	I
1	5	7	1	346	368	2	468	68	9		Step 14: Determine all possible # combinations for vacant cells:
2	2	48	6	9	18	5	3	18	7
3	48	9	3	7	168	468	2	5	1468		Obvious observations:  #5 in D5; #8 in F5
4	7	5	4	136	2	368	68	9	68
5	6	3	9	5	4	8	1	7	2		Step 15: re Block EE: #1 only shows up in Cell D4…
6	1	2	8	56	7	9	456	3	456
7	348	1	5	346	36	7	9	2	368
8	39	6	2	8	359	1	7	4	35
9	3489	48	7	2	3569	346	568	168	13568
	A	B	C	D	E	F	G	H	I
1	5	7	1	346	368	2	468	68	9		Step 16: Determine all possible # combinations for vacant cells:
2	2	48	6	9	18	5	3	18	7
3	48	9	3	7	168	46	2	5	1468		Obvious observations:  #6 in D6 (& therefore #3 in F4)
4	7	5	4	1	2	36	68	9	68
5	6	3	9	5	4	8	1	7	2		Step 17: re Row 2: #4 only shows up in Cell B2
6	1	2	8	6	7	9	456	3	456
7	348	1	5	346	36	7	9	2	368
8	39	6	2	8	359	1	7	4	35
9	3489	48	7	2	3569	346	568	168	13568
	A	B	C	D	E	F	G	H	I
1	5	7	1			2			9		Step 18: Complete Block AA: #8 MUST go in Cell A3...
2	2	4	6	9		5	3		7
3	8	9	3	7			2	5			Step 19: #8 is now in Columns A & C: #8 MUST go in Cell B9...
4	7	5	4	1	2	3	8	9
5	6	3	9	5	4	8	1	7	2		Step 20: #8 is now in Rows 8 & 9: #8 MUST go in Cell I7...
6	1	2	8	6	7	9		3
7		1	5			7	9	2	8		Step 21: #8 is in Rows 5 & 6: #8 MUST go in Cell G4
8		6	2	8		1	7	4			(#8 from Step 20 @ I8 excludes use of Cell I4)
9		8	7	2
	A	B	C	D	E	F	G	H	I
1	5	7	1			2			9		Step 22: Complete Row 4: #6 MUST go in Cell I4...
2	2	4	6	9		5	3		7
3	8	9	3	7			2	5
4	7	5	4	1	2	3	8	9	6
5	6	3	9	5	4	8	1	7	2
6	1	2	8	6	7	9		3
7		1	5			7	9	2	8
8		6	2	8		1	7	4
9		8	7	2
	A	B	C	D	E	F	G	H	I
1	5	7	1	34	368	2	46	68	9		Step 23: Determine all possible # combinations for vacant cells:
2	2	4	6	9	18	5	3	18	7
3	8	9	3	7	16	46	2	5	14		Obvious observations:
4	7	5	4	1	2	3	8	9	6
5	6	3	9	5	4	8	1	7	2		Step 24: re Row 2: # only shows up in Cell
6	1	2	8	6	7	9	45	3	45
7	34	1	5	34	36	7	9	2	8
8	39	6	2	8	359	1	7	4	35
9	349	8	7	2	3569	46	56	16	135
	A	B	C	D	E	F	G	H	I
1	5	7	1	34	368	2	46	68	9		Step 25: Determine all possible # combinations for vacant cells:
2	2	4	6	9	18	5	3	18	7
3	8	9	3	7	16	46	2	5	14		Obvious observations:  None!
4	7	5	4	1	2	3	8	9	6
5	6	3	9	5	4	8	1	7	2		Step 26: re Row 7: #6 only shows up in Cell E7
6	1	2	8	6	7	9	45	3	45
7	34	1	5	34	36	7	9	2	8
8	39	6	2	8	359	1	7	4	35
9	349	8	7	2	3569	46	56	16	135
	A	B	C	D	E	F	G	H	I
1	5	7	1	4		2			9		Step 27: #6 is in Columns D & E: #6 MUST go in Cell F3...
2	2	4	6	9		5	3		7
3	8	9	3	7		6	2	5	4		Step 28: Block BB needs #4: #4 MUST go in Cell D1
4	7	5	4	1	2	3	8	9	6		(#4 in Column E @ E5 exludes use of Cells E1, E2 & E3)
5	6	3	9	5	4	8	1	7	2
6	1	2	8	6	7	9	4	3			Step 29: #4 is now in Rows 1 & 2: #4 MUST go in Cell I3...
7		1	5		6	7	9	2	8
8		6	2	8		1	7	4			Step 30: #4 is now in Columns H & I: #4 MUST go in Cell G6...
9		8	7	2
	A	B	C	D	E	F	G	H	I
1	5	7	1	4		2			9		Step 31: Complete Column D: #3 MUST go in Cell D7...
2	2	4	6	9		5	3		7
3	8	9	3	7		6	2	5	4		Step 32: Complete Block FF: #5 MUST go in Cell I6...
4	7	5	4	1	2	3	8	9	6
5	6	3	9	5	4	8	1	7	2		Step 33: #5 is now in Columns H & I: #5 MUST go in Cell G9...
6	1	2	8	6	7	9	4	3	5
7		1	5	3	6	7	9	2	8		Step 34: #5 is now in Rows 7 & 9: #5 MUST go in Cell E8...
8		6	2	8	5	1	7	4
9		8	7	2			5
	A	B	C	D	E	F	G	H	I
1	5	7	1	4		2			9		Step 35: Complete Block HH: need #s 4 & 9…
2	2	4	6	9		5	3		7		#9 can't go in Cell F9 (cf #9 in Column F @ F6), so
3	8	9	3	7		6	2	5	4		#9 MUST go in Cell E9,  therefore #4 MUST go in Cell F9
4	7	5	4	1	2	3	8	9	6
5	6	3	9	5	4	8	1	7	2		Step 36: #4 is now in Rows 8 & 9: #4 MUST go in Cell A7...
6	1	2	8	6	7	9	4	3	5
7	4	1	5	3	6	7	9	2	8		Step 37: #9 is now in Rows 7 & 9: #9 MUST go in Cell A8...
8	9	6	2	8	5	1	7	4
9	3	8	7	2	9	4	5				Step 38: Complete Block GG: #3 MUST go in Cell A9...
	A	B	C	D	E	F	G	H	I
1	5	7	1	4		2	6		9		Step 39: #3 is in Rows 7 & 9: #3 MUST go in Cell I8...
2	2	4	6	9		5	3		7
3	8	9	3	7	1	6	2	5	4		Step 40: Complete Column G: #6 MUST go in Cell G1...
4	7	5	4	1	2	3	8	9	6
5	6	3	9	5	4	8	1	7	2		Step 41: Complete Row 3: #1 MUST go in Cell E3...
6	1	2	8	6	7	9	4	3	5
7	4	1	5	3	6	7	9	2	8
8	9	6	2	8	5	1	7	4	3
9	3	8	7	2	9	4	5
	A	B	C	D	E	F	G	H	I
1	5	7	1	4	3	2	6		9		Step 42: Complete Block II: need #s 1 & 6…
2	2	4	6	9	8	5	3		7		#6 can't go in Cell I9 (cf #6 in Column I @ I4), so
3	8	9	3	7	1	6	2	5	4		#6 MUST go in Cell H9, & therefore #1 MUST go in Cell I9
4	7	5	4	1	2	3	8	9	6
5	6	3	9	5	4	8	1	7	2		Step 43: Complete Block BB: need #s 3 & 8…
6	1	2	8	6	7	9	4	3	5		#3 can't go in Cell E2 (cf #3 in Row 2 @ G2), so
7	4	1	5	3	6	7	9	2	8		#3 MUST go in Cell E1, & therefore #8 MUST go in Cell E2
8	9	6	2	8	5	1	7	4	3
9	3	8	7	2	9	4	5	6	1
	A	B	C	D	E	F	G	H	I
1	5	7	1	4	3	2	6	8	9		Step 44: Complete Block CC (& Puzzle): need #s 1 & 8…
2	2	4	6	9	8	5	3	1	7		#8 can't go in Cell H2 (cf #8 in Row 2 @ E2), so
3	8	9	3	7	1	6	2	5	4		#8 MUST go in Cell H1, & therefore #1 MUST go in Cell H2
4	7	5	4	1	2	3	8	9	6
5	6	3	9	5	4	8	1	7	2
6	1	2	8	6	7	9	4	3	5
7	4	1	5	3	6	7	9	2	8
8	9	6	2	8	5	1	7	4	3
9	3	8	7	2	9	4	5	6	1
	A	B	C	D	E	F	G	H	I
1	5	7	1	4	3	2	6	8	9	45	Final Proof:
2	2	4	6	9	8	5	3	1	7	45	- All Rows add up to 45;
3	8	9	3	7	1	6	2	5	4	45
4	7	5	4	1	2	3	8	9	6	45
5	6	3	9	5	4	8	1	7	2	45
6	1	2	8	6	7	9	4	3	5	45
7	4	1	5	3	6	7	9	2	8	45
8	9	6	2	8	5	1	7	4	3	45
9	3	8	7	2	9	4	5	6	1	45
	45	45	45	45	45	45	45	45	45		- All Columns add up to 45;
	45			45			45				- All 3x3 Blocks add up to 45…
	45			45			45
	45			45			45